Make games go faster ideas
If the system uses initiative, explicitly tell your players to not post in initiative order. Have them post as soon as they can and you as DM adjudicate the round in initiative order. Some granularity will be lost and some actions won’t make sense as you work through the round. Those can be missed turns, attacking an already dead opponent, whatever. Just run with it.
Set time limits on posting or posting requirements, be explicit about them, and skip players who don’t post within that framework. It sounds harsh, but waiting for everyone drags out the game and increases the chance people will disappear. It’s not a punishment, it’s to keep the game from dying.
Making sure to include everyone even if the player can’t post. In combat you can have players set default actions. The rogue will always try to hide and get advantage on an attack. The barbarian will rage and charge. The wizard will blast with a cantrip. Etc. That works well enough, but OOC is tricky and players tend not to like having their characters god-modded.
Simple vs complex systems. Just have your players roll everything they need to resolve an action with their initial post and you can generally run any system just fine. Take D&D as an example. To attack you need to roll to-hit, find out if that hits, then roll damage, then the DM narrates the outcome. You can speed up play by just having the players roll to-hit and damage when they make their post declaring the attack. Similarly, most players are going to be familiar with the basics of the games they play, so if they perform an action that would require a roll, have them make those upfront, like the to-hit and damage rolls in combat. If they try to sneak they should include a stealth roll just in case. If they try to climb something they should make an athletics roll just in case. Etc. Be clear that the roll isn’t binding and that it’s a precaution. Like if it’s impossible and they roll a 20, it doesn’t matter; likewise, if you wouldn’t have required a roll and they roll a 1 don’t punish them for it.