Can't go over everything but, these bits stood out:
I see. A player rolling to hit twice in one round slows things down. What about a spell that forces saves for half damage from multiple targets?
Yes. So do interrupt attacks, conditional stuff, complex modifiers, etc.
By focusing on DPS (or effectiveness per second) instead of DPR (or effectiveness per round), you can prevent fights that take extra
table time.
And spells quite often take more time to resolve than attacks; the time it takes should be carefully bounded to the scale of how big an impact it has on the combat.
If your game is going to embrace the "full adventuring day", then you can even weigh this over the adventuring day. If wizards have simple to resolve cantrips and then a few more complex to resolve spells that have bigger impact, and fighters have medium-complex to resolve weapon attacks at will, the trade off of damage per second (or effectiveness per second) can occur between encounters not just between rounds.
So, make an attack roll on one round, on the next round force multiple saves and make another attack roll - will speed things up?
I'm not sure that, if you, say, did both of those, it'd have a big impact on the turn cycle. Maybe if you had more characters just making an attack per round than you did casters?
Different tables can resolve turns much faster or slower than others, and different individuals have vastly different perceptions of time and tolerance for 'delay' (ie, getting the action back to the only turn that matters, theirs.) Of course, if a game is balanced, players are comparably skilled, and play time is fairly distributed, then, in oder to get back to your turn quickly, you turn will aslo be shorter. That's the best 'fast combat' can hope to accomplish - cycling very short turns. Whether you get 12 turns that each resolve in 30 sec or 3 turns of 2 min each out of an hour of play, you're still not taking your turn for 54 min of that hour.
Sure; but what I find is often the case is that the mechanics of one set of options are way out of kilter. Like, you use option X, and your DPR goes up but your DPS plummets.
It is less a problem in 5e than in other games. But an example of a good DPR but poor DPS spell is animate objects with 10 tiny objects at a table where the DM gives the control to the PC. Resolving 10 attacks for 6.5 damage each and moving 10 creatures around is going to consume time, with not that high of a damage per second of time spent on the actions.
In comparison, a spell like disintegrate. It does one roll to do ~75 damage, or not (40+10d6). The damage per round of animate objects matches disintegrate (ok, a bit lower, as disintegrate might be more accurate), but per second that disintegrate blows animate objects out of the water.
Auras that do 5 damage to adjacent foes are low damage per second in my experience, but easily great damage per round if they are passive.
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And while some players really care about how much time they spend, often they care a bunch about the size of the impact their actions have on the game narrative. Like, a game that had 10 minute turns, 4 players, and 3 rounds to finish a fight that takes 2 hours, and the PCs are in peril the entire time. Another game with 1 minute turns, 4 players, and 6 rounds to finish a fight will be over in half an hour; if it has the same feelings of peril...
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So, game mechanics that permit damage dice to be rolled with attack dice? Improves damage per second. The more conditional chains you have, the slower the gameplay.
OTOH, keeping things active is also a great thing. Interrupts, where you get to do things on other player's turns, keep you engaged - but they also significantly slow down play in my experience. And if your power budget is spread over an interrupt and your action, your damage per second drops.
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I'm not saying this is the solution to everything. I'm just saying distinguishing between damage per round and damage per second is a valuable thing to do at a table. I personally found using it made it extremely clear that some mechanics where not worth the table top time price when making characters. And when making monsters, it helps keep my eye on the ball to keep complex mechanics on high-octane abilities only.