I understand that the design was attempting to be different than the Valor Bard and Bladesinger, but I find it a misstep to want to shy away from magical items that is typically associated with martial combatants when designing a spellcaster subclass that wants to simulate a martial. You still have some magical items that can improve your spell attack, like the Wand of the War Mage or the Bloodwell Vial and of course a plethora of magical staffs and assorted high-end robes. It just feels a bit weird or perhaps anticlimactic that holding a wand makes you better at hitting with a blade made of magical flames.

Regarding the concentration-free Flame Blade, I would have expected that for a design that relies so heavily on the Flame Blade spell to live out its fantasy of emulating a martial. The amount of friction caused by the 2024 Ranger having free Hunter's Mark and class features to build on top of it, and still have it require concentration whilst there are other Ranger-specific spells that require Concentration and other features in the class and for other classes that specifically removes Concentration from subclass specialized spells, that would seem like a bad pitfall to fall into.
Having said all that, I don't think Flame Blade have Concentration as a balance measure, I believe it is more to be consistent with other similar types of spells. So removing it doesn't really move the needle by much. Secondly the Sorc don't have access to CME, so they will mostly use their concentration for Greater Invisibility or Blur to increase their martial performance. Haste doesn't really make much sense when they cannot take the Magic action to attack with the Flame Blade and they are unlikely to have a secondary weapon on hand. More likely they would just concentration on one of the prime control/debuff/support spells available to them (Hypnotic Pattern, Banishment, Wall of Stone, Reverse Gravity, Hold Monster - or to Haste a better Martial).

You cannot cast spells at 10th level, there is no mechanical backing in the rules. Even the Sorc's Twinned Spell Metamagic only says it increases the spell's effective level by one, it doesn't cast as that higher level. There's general upcasting rules, so a GM could skirt the issue but an official piece of content that breaks the general rules without specifically saying it can break the rules, is prone to cause confusion and not be a good design. I'm not sure how to fix this without saying you can break the rules, except to always cast Flame Blade at 2nd level when using Sorc Points and then have Arcane Blade's feature change its effective level to the amount of Sorc Points you spend. This has the probable unintended consequence that for contests (like Dispel Magic or 2014-Counterspell) or suppression-effects that cares about spell level, they would have an easier time to ruin your fun.
Another small issue I have with Arcane Blade's Sorc Point avenue is that it doesn't specify that you have to use at least 2 Sorc Points - since you cannot downcast a spell but nothing in the rules prevents me from assuming I can spend 1 Sorc Point and cast FB at level 1. Easy fix to include a minimum of 2 in the text for spending Sorc Points to cast at that level.

Again Mage Armor + Shield nets you 13 + 2-3 DEX + 5 from Shield. 15-16 AC base is going to be a problem going into high tier 2-play and you wont improve it most likely, except with maybe Bracers of Defense and a Cloak of Protection. That's two magic items you cannot guarantee to acquire for a campaign, nor is it particularly satisfying to rely on specific magic items for the subclass to be functional in melee range. You're also dedicating your Reaction to using Shield on top of basically all your lower level spell slots.

A bit of personal experience playing a Bladesinger with Mage Armor's 13 + 3 DEX (because I had CON saves covered) + 5 from Bladesong + 1 Cloak of Protection. I often still had to cast Shield in tier 3+ play and it was not always enough. That's +5 more than the Sorc you're proposing - at least until level 14 where a free Blade Ward [avg. 2.5 AC] is added atop casting Shield. And while the Sorc have a bunch of Temp HP - those disappear very rapidly when your AC don't keep you from getting hit, and your Reaction is mostly locked to cast Shield (to have similar AC as other Martials in high tier 2 and beyond play), so no Absorb Elements/Silvery Barbs/Counterspell.

I see issues playing this subclass as a martial without multiclassing to fix my AC situation. Another thing is that I don't know if the benefits to going melee with Flame Blade is all that better from just staying on the backline with a bunch of bonus Temp HP and slinging my spell slots to deal damage or provide control/utility. Not that this is significantly different from the Bladesinger who could just as easily take the subclass' defensive tools and remain a ranged caster. But at least on the Bladesinger I don't feel scared going into a fight, nor do I feel like I have to spend an enormous amount of resources to barely feel like a martial.

Oh, I didn't realize this before but I fail to see any value in the Arcane Blade benefit that lets your Flame Blade count as a weapon worth 10 GP. It doesn't specify that the Flame Blade is any type of weapon nor the statistics for it - is it a short sword, club, crossbow? So even if it technically satisfies the requirement to cast True Strike (or similar) it is not a weapon for which you can attack with. The Flame Blade spell specifically instructs you to make a spell attack using a Magic action to deal its damage. So there's no synergy with Sword-cantrips.
Just thought to point it out and that's a reason to make the Arcane Blade specify a weapon type that it emulates. It doesn't functionally change anything about the Flame Blade's modified damage, it just adds the option to attack with the emulated - whatever - short sword, and THAT can be used with True Strike, Booming Blade, or Green-Flame Blade.

It’s a full spellcaster. It shouldn’t also be a full martial. Well if you want a full caster that is also a full martial you need a new class. It can’t be balanced in the subclass.

I also used Flame Blade to lower the amount of words I had to type. I could have just typed out a feature that lets you summon a blade of arcane energy. Then it would need ways to interact with other sorcerer features like metamagic, so more words to type. Also flame Blade was used to balance certain things. Self Haste is a bad play style in my opinion of a team game. I wanted to avoid that. 

It’s not about casting spells at 10th level. The game already supports what happens if you upcast a spell at ninth level and have a feature that cast is one level higher. You increase the damage by whatever it says, or it targets more creatures. Also it doesn’t matter because I wouldn’t design the class around flame blade if I did it again.

As far as having to cast shield a lot while in close combat I say, “good.” This type of design isn’t meant to always be a martial, it is a spellcaster.

Again I say, “good,” if you need to multiclass to be a full martial that is acceptable. It’s what should happen. 

I would change so much about this now, it’s not even worth trying to fix the problems in this design. The goal of my design doesn’t align with what most players actually want, and as you point out it has a bunch of things that literally don’t work. I did make it in a rush. I’m a through spaghetti at the wall kind of guy, so I’ve already come up with two designs that are closer to what people would actually want, but every Gish is going to fall short because what people really want is a gestalt character. 

A design that draws from actual D&D history. It is a Gish, but it is stilled flawed. Spend too much time in melee and you will die. Level 18 feature needs a lot more words to explain exactly what spells are allowed to be used. 

Duskblade Sorcery

3rd: Duskbalde Spells

• 3rd: Faerie Fire, Thunderwave, Hold Person, Ray of Enfeeblement
• 5th: Counter Spell, Fireball
• 7th: Blight, Victriolic Sphere
• 9th: Hold Monster, Synaptic Static

3rd: Arcane Channeling

When you cast a spell from the Duskblade Spell list on a target in range of a melee weapon you are holding, you can deliver the spell by making an attack roll with the weapon. If the attack hits, the target takes the weapon damage and then the spell effect is resolved as if the target failed the saving throw. If the spell delivered with this method normally affects an area or could target multiple targets it does not. Only the target hit with the melee weapon attack is affected. If the attack misses the spell slot is still expended as the spell has no effect, but your next attack with the weapon that hits before the end of your next turn deals an additional 1d6 per spell level force damage.

3rd: Close Combat Mage

You have proficiency with Martial Weapons and can use your Charisma modifier for attack and damage rolls of melee weapons. You can use Simple and Martial melee weapons as a spell focus. You have training with light armor. If you are hit with an attack while wearing light armor you can use your reaction to reinforce the armor adding your Charisma modifier to your AC until the start of your next turn possibly causing the attack to miss. You can’t reinforce your armor this way until you use a bonus action to reset your armor, complete a short or long rest, or    doff and don the same or new armor. 

6th: Follow Up

When you take the Magic Action you may make a melee weapon attack as a bonus action. When you take the Attack Action you may cast a cantrip as a bonus action.

14th: Extra Attack

You can attack twice instead of once when you take the attack action. 

18th: Improve Channeling

You can use Arcane Channelling to deliver any spell you have prepared that is one action and requires a saving throw.

This one was actually made first and focuses on being able to survive in melee when you use Innate Sorcery. Definitely has flaws considering I spent less time on it. It has ideas I think are fun. 
Battle Sorcery

3rd Battle Spells

• 3rd: Armor of Agathys, Mage Armor, Arcane Vigor, Magic Weapon
• 5th: Conjure Barrage, Wind Wall
• 7th: Death Ward, Fire Shield
• 9th: Steel Wind Strike, Wall of Force

3rd Blade Magic

When you wield a magical melee weapon you gain proficiency with the weapon until you are no longer holding it. The weapon can be one that is permanently magical or one that is temporarily magical due to a spell like Magic Weapon or Light (cantrip) being cast upon it. Additionally you may use your Charisma modifier for Attack and Damage rolls of any melee weapon you are proficient with. 

6th Innate Combat Magics

While using Innate Sorcery you gain the following magical benefits. 

• Defensive Shroud- You gain a bonus to your AC equal to half your Charisma modifier rounded up if you aren’t wearing armor or using a shield (minimum +1)
• Fast Strikes- If you have not cast a spell using a spell slot this turn you can take a Magic Action to make two weapon attacks with a magical weapon. You can not cast a spell using a spell slot until the end of this turn. 
• Spell Blitz- Immediately after a creature you can see within 100ft  takes damage from a 1st level+ spell you cast this turn, you may use your bonus action to teleport to an unoccupied space within 5ft of the creature and make one melee weapon attack against it. 

14th Explosive Strike

Once on your turn when you hit with a weapon attack you may expend a spellslot to cause your weapon to erupt with magical energy sending a 30 ft line or 15ft cone straight through the target dealing 1d8+ 1d4 per spellslot level Acid (1), Cold (2), Fire (3), Lightning (4), Necrotic (5), Poison (6), Radiant (7), or Thunder (8), Psychic (9), Force (10) damage to the target and any other creature in the area that fails Dex save against your spell save DC. The damage type depends on the roll of the d10.

18th Battle Magic Mastery

You gain the following benefits.

• Extra Attack- You can attack twice instead once when you take the attack action.

• Defensive Cloud- You always have the bonus to AC from Defensive Should, when using Innate Sorcery it increases to your Charisma modifier instead of half rounded up.
• Faster Strikes- You can make three melee weapon attacks with Swift Strikes instead of two. 

• Improve Spell Blitz- When you use Spell Blitz you can make two weapon attacks as part of the bonus action instead of one.

• Precision Explosive Strike- You can choose the damage of Explosive Strikes instead of letting the d10 decide.