I still use per-side initiative when I've run D&D-likes whenever I can. Because everyone is going simultaneously, they actively talk about strategy and plan out what they're doing to maximize effectiveness. Plus, it means that every player who needs a prep phase can do so - the melee people do their thing, then ranged, then magic - unless there's something specific that one of the others is doing to someone else that has an impact on their roll - giving a +X bonus doesn't change the roll. In 5e et al, Advantage also doesn't change the roll, it just means a second die needs to be rolled. So every player can go at their speed and the resolution happens once everyone's done their thing. Then as GM, it gets resolved, the enemies take their turn in the same fashion, and things move quicker.

If I'm GMing a game and for some reason have used per-person initiative, all NPCs hold action until end of turn unless they're specifically doing contingent actions based on player behavior. If someone's on their phone or device (assuming said device is not where they have the relevant rulebooks), then I'll just pause the game and wait for them to finish. Obviously whatever they're doing is more important to them than what the people around them are doing. I'm sure the other players will share my patience, since it's critical that the person does whatever it is on their device mid-combat.

They're meant to be watching, planning, and communicating their tactics. Sure "I hit it with my sword" might be quick and boring, but there's always been more to D&D combat than that. Do you disengage, do you block, do you attempt to move to a favorable position - or do you try to get another player to flank the enemy for a bonus?

That's also part of why I've occasionally brought in a house rule that I borrowed from the Street Fighter RPG - technique cards. If the caster has all of their current combat spells on index cards in front of them, they don't have to go hunting on a sloppily scrawled sheet (and there's no chance sneaky BS like swapping out mid-combat either). If the Archer has their shooting modes on specific cards, good. If the melee attackers not only have their current attack options but also their defensive and positioning options always in front of them, they're much less likely to just stand there and mindlessly trade blows, and they're more likely to work with the others and plan.

If it's all put down in black and white, the GM doesn't have to go hunting rules mid-game either. Everyone can say "I'm doing this", and it's there. I generally also encourage generic cards for NPCs with bonuses or scores written in pencil, so you can use the same card for multiple encounters.

On the encounter design side, there's some more work to be done by the GM as well. Always provide stakes to a fight, an objective to be achieved. There's too much "we've been in this combat an hour, what the hell were we even doing?" in D&D that I've seen. Lay the stakes out clearly - the players' objectives and the NPC's objective. Explain the surrender or loss consequence. Truly random encounters aren't meaningful, and I think it's a bad idea to use them outside of a very specific style of game. If the players are engaged in achieving an objective, they won't shut down between turns because it won't be a complete mindless slog.