I once again want to state how much I appreciate every piece of feedback I have received so far.

I have just updated my subclass in the original post (with the original version placed in a spoiler tag for comparison). While I have not addressed every response I received, I have updated a few things based on the responses so far

1) Change to Step of the Wind/Patient Defense sections of Primordial Martial Arts: I moved away from giving the monk even more extra speed, but still wanted something which could easily scale up with spell slot level. For this, I am trying to take inspiration from the UA spell Flame Stride which allows a creature to deal passive damage. This effect I think will work very well with a monk's high mobility if it finds itself weaving through a large group of enemies.

Changed the Patient Defense benefit to being a flat damage reduction (like Heavy Armor Master) which can scale with spell slot level. Chose to do this so that not every part of Primordial Martial Arts was dealing damage. This gives the monk a passive damage reduction for turns where it may be targeted by several attacks on top of the higher chances of avoiding being hit altogether with dodging

2) Change to Earth section of Four Element Stance: Now extends the benefits of Evasion to both Str and Con saving throws, making the monk have a strong defense against physical effects

3) Change to Air section of Four Element Stance: Now allows you to gain the benefits of both dashing and disengaging as part of the same bonus action taken with Step of the Wind.

4) Change to spellcasting benefit of Four Element Stance: Now allows you to cast four 5th level spells, once each, while the feature is active. These spells require no spell slots, but do count as being cast with a 5th level spell slot for interacting with Primordial Martial Arts and Metamagic.

Don't forget to add the line: "These count as Monk spells for you" when taking spells from any list.

As for the type of casting I'd suggest making the Four Elements a 1/2 caster rather than a 1/3 caster, but like Artificer give them cantrips as well.

That would give them access to spells at roughly the same rate as Four Elements do now, maybe a bit faster, receiving third level spells like fireball and fly at 9th rather than 11th level. That would also allow them to access up to level five spells eventually at level 17. I think you may have to make your own spell list for the Monk, since just accessing everyone's elemental spells might be a bit too strong. If you do that you can make them like a Paladin, able to select half their Monk level plus their wisdom modifier in spells from their list each day. Spells like "Cure Wounds" should be in there too.

Something like:

Spell list:
Cantrips:
Booming Blade
Control Flames
Create Bonfire
Fire Bolt
Frostbite
Green-Flame Blade
Guidance
Gust
Lightning Lure
Mold Earth
Prestidigitation
Produce Flame
Ray of Frost
Shape Water
Shocking Grasp
Thorn Whip
Thunderclap

And so on up to level 5. Basically spells which invoke the elements, and occasional healing spells or utility (like Fly).

Quote from Freman_Bloodglaive >>

Don't forget to add the line: "These count as Monk spells for you" when taking spells from any list.

As for the type of casting I'd suggest making the Four Elements a 1/2 caster rather than a 1/3 caster, but like Artificer give them cantrips as well.

That would give them access to spells at roughly the same rate as Four Elements do now, maybe a bit faster, receiving third level spells like fireball and fly at 9th rather than 11th level. That would also allow them to access up to level five spells eventually at level 17. I think you may have to make your own spell list for the Monk, since just accessing everyone's elemental spells might be a bit too strong. If you do that you can make them like a Paladin, able to select half their Monk level plus their wisdom modifier in spells from their list each day. Spells like "Cure Wounds" should be in there too.

Something like:

Spell list:
Cantrips:
Booming Blade
Control Flames
Create Bonfire
Fire Bolt
Frostbite
Green-Flame Blade
Guidance
Gust
Lightning Lure
Mold Earth
Prestidigitation
Produce Flame
Ray of Frost
Shape Water
Shocking Grasp
Thorn Whip
Thunderclap

And so on up to level 5. Basically spells which invoke the elements, and occasional healing spells or utility (like Fly).

 I do not want to make the monk subclass a half caster. At least in 5e so far, being a half-caster is baked into the main body of a class. When casting is introduced as part of a subclass, you generally get 1/3 caster progression (with the exceptions to this being the original 4EM's ki method and the (unofficial) bloodhunter's quasi-warlock subclass with pact slots). In any case, no subclass gives the benefits of half or full-caster progression.

Also, worth mentioning that this monk cannot access everyone's spell list. It specifically draws from Druid and Sorcerer. Defining what type of spells you can select based on the nature attribute leaves it open ended for when more spells are added to the game. While the number of qualifying spells that already exist from both spell lists isnt huge, it is extensive enough where writing up a full spell list for the subclass would be a lengthy addition.

I still appreciate the idea though.

Notes on changes 10-7-21

Spellcasting:  Changed language for Cantrips to restrict choice to just those from Druid & Sorcerer spell list (original intent)

Ki Empowered Casting: Changed number of metamagic options known at 3rd level to two, with a third option gained at 17th level. Changed language to allow switching metamagic options out each time you gain a level which grants an ASI, rather than every level

Primordial Martial Arts: Removed use of Flurry of Blows and changed bonus to be ability to make two unarmed strikes after taking Cast a Spell action by spending 1 ki (effectively the same thing, but without possible conflict arising from language of Flurry of Blows).

Changed bonus damage for unarmed strikes to be flat equal to twice level of spell slot. This brings the bonus damage from this feature per long rest to be 56 (if the monk lands one of these attacks on average) or max 112 (if it lands both) [for a 20th level monk]. The original 1d6 allowed for a possibility of max damage of 300+ per long rest, which feels like too much

Four Element Stance: Changed to only allow casting one of the listed spell options, rather than all four. I still want to include 5th level spells as part of the capstone, but it would feel too strange to allow the monk to cast four 5th level spells 1/day as part of the capstone when at level 17 they cant normally even cast a 4th level spell using their Spellcasting feature. A single 5th level spell is still a little weird, but not enough to merit change in my opinion. 

Changed out Cone of Cold for Maelstrom and Flame Strike for Immolation. The goal is for all the capstone spells to be ongoing, concentration spells that can potentially persist for the full duration of Four Element Stance rather than instantaneous effects.  Removing Flame Strike also removes issue of having a spell which deals radiant damage, which is out of theme for this monk