So we're attempting to save time on rolls by rolling to hit, then adding anything over AC to damage.  An example:

I roll a 16 to hit basic human commoner, basic human commoner's AC might be 12.  So I've rolled 4 points over their AC.

Now I roll damage, so I do 2d6+4, since I rolled 4 over their AC originally.

If my weapon already adds damage, I disregard this.

So my +1 sword might do, +1 to hit, I roll 16, get a total of 17 against basic human, AC 12, I've rolled 5 over.

I roll damage ignoring the +1 to damage because I've already included it in my to hit roll. so instead of 2d6+1+5, it's just +5.

------------------------------

Not only does this seem unfair to me, especially if I roll exact AC, but I've crunched the numbers on some larger creatures and we're absolutely going to get creamed down the line against creatures doing +19 to hit, etc.

Can you guys help me come up with a compelling argument to present that saves feelings but convinces everyone that we're only hurting ourselves and not really saving any time?

It doesn't save time. It's silly and not a good way to go about it. 

2 options. 

Roll hit dice.. does it hit? Yes then roll damage. 

Roll hit dice and damage dice same time. 

The rules as written RAW. Are there for a reason. 

My sword says that if I hit then I get to roll 2d6+2 and I'm going to roll that and the enemy is going to take that much damage. Period. 

The most compelling argument to me is that this isn't how attack rolls and damage are designed to work. Attack rolls (d20 + ability + proficiency) are designed to beat AC, a binary yes or no result. This is how the AC of monsters is determined, based on the likelihood of players either succeeding or failing.

Weapons and hp however are designed with each other in mind; if attack/to hit is the success/failure metric, weapon damage/hp is the clock mechanic, deciding how long a fight goes on for. Again, monster hp is designed with average damage output in mind.

Also, there's the fact that this change to the rules makes every weapon idenitical, with the only trait that matters is finesse and two-handed being nothing but pure downside. There's no reason for every PC to just stock up on daggers; they now not only do the same amount of damage as every other weapon, but also are the lightest and can be thrown. This house rule literally removes the need to use anything else.

This totally imbalances the importance of the combination of high attack bonus and high AC, as Davedamon explains. DEX is now even more important with finesse weapons, while lower AC / higher HP builds are devalued. 

Saving throw spells with damage are getting devalued in comparison to potential extra damage for spell attacks.

Overall, this looks like breaking a lot of things, while achieving not much.

How is he running it on attack spells?

A silly change IMO that definitely doesn’t speed things up. 

Yeah, this is a horrible idea that doesn't do anything to speed things up. 

The only easy way to speed things up (IMO) is the method Avohei mentioned. Roll the damage die and the attack die at the same time. For example, attacking with a dagger. Roll a d20 and a d4 and it will go something like this:
"I rolled a 17 to hit and if that hits the enemy takes 5 piercing damage from my dagger."
Quick and easy.

spell attack is the same, anything over ac is added

of course if a spell specifically says add spell level or another d6 we add those

this mostly screws weapon users, and while it has been helpful at lower levels when things have lower AC I foresee it getting bad around level 8

Yeah this is a bad idea. When you change a core mechanic like this, you effect everything that hangs off of it. It messes with a lot of the assumptions made for other mechanics and I agree that it doesn't even really speed up the game at all.

There are lots of ways to shorten combat.

• lower enemy HP and increase their damage
• have enemies surrender/flee/bargain when they have clearly lost or just narrate the mop-up phase of the fight
• have alternative win conditions that don't require reducing every enemy in the room to 0 hp
• roll attack and damage together, use an app to add it all immediately (10d6 takes some time for anyone to add up)
• every turn call out the player "on deck" who will go next so they can begin planning their actions

Often if you actually watch during combat, it's decision-making and attention issues that plague most slow tables. Some players may benefit from a flow-chart that helps choose their actions based on obvious YES/NO questions like "is anything adjacent to me?" or "does someone need healing?" That and getting people to put their phones away can do a lot.

Additional detractions:

• Now once someone hits, everyone knows the opponent's AC. I find it pretty rare for a DM to want to give away the NPCs' stats, as it adds to tension and party communication/strategizing. It sounds like your DM is fine with it, but pointing this out might help entice them as something useful to them.
• This method, being completely different from regular gameplay, is likely to slow down seasoned players as they have to switch mental gears and redo maths.
  • It'll also confuse new players if they learn this method and then go on to other tables, for the same reason. Even if they know that this table's dice rolling was unique, it's still establishing a mental pattern.