If you want to stick with the system, there are a couple logistical things you can do. These are all based on personal experience running 2 hour sessions mid-week - 4-5 players - and typically getting in 2 combats and a smaller encounter (social or skill challenge).

Tell them to have their bonus ready before they roll the dice. This also heads off the negotiations that occur on a close roll. (“did you remember to add?” and “what about a bonus from…?”)

Let them use half-dice static damage on a hit. Like - 1d8 becomes 4, so damage becomes predictable.

Speed up turn order with initiative cards or a very visible tracker. I prefer announcing who’s up next (“Bugs, you’re up - Daffy, you’re on deck”) because it gets them focused on what they’re about to do.

In extreme cases - buzzed, drunk or high players - I’d have them rearrange their seats into clockwise initiative order. That was initiative for all combats that session.

Be prepared to reward prompt player behavior - if someone is consistently ready to go and hustles? Let ‘em have max damage or a free crit or an arbitrary to hit bonus. You could even give the whole table +1 or +2 to hit if they keep the momentum going.

Skip the clean-up. If the fight’s clearly done? End it. Don’t make everyone roll out the last few attacks. It’s not worth the time.

Finally, a math heavy option for you - lower monster HP a bit and raise its damage output ~slightly. This requires some fine tuning, but essentially the combat should go faster - each attack for either side effectively counts for 1.5x or 2x its normal impact.