This does sound very cool - I really don't like the "combat swoop" that rolling initiative introduces. Everything slows down as we methodically make our way through each persons turn. This sounds like "narrative combat" and it's something I plan to do when combat ensues with an enemy that is vastly mismatched with the PCs (hasn't come up yet). But the thought of doing it all the time is appealing.
How do you ensure that everyone gets a chance and it just doesn't turn into a free-for-all (and the more hesitant players aren't just rolled over)?
We still have rounds, so everybody gets to act each round, it's just that we don't manage all of the actions sequentially with initiative. The way it usually works is that people will just start telling me what they plan to do, if they have questions about what the monsters are doing I'll answer them, and other people will jump in. If I don't hear somebody, I just ask. Tactics get worked out in the process, and they can roll any time they'd like after they've declared their action, and we just resolve them at the end of the round. Some things, like casting spells, take more time than others. This can show up as a modifier to an initiative check if needed, but usually isn't.
The majority of time, when something occurs isn't really relevant. But if you need to know whether the rogue killed the orc before the orc hit the wizard, then you make an opposed initiative check. This is particularly important in our campaign, because you can interrupt spellcasting (a la AD&D), although it doesn't automatically cause failure.
Part of why I love it is that it is so chaotic, as combat should be. I've run it at a table with 12 players, and people figured out pretty quick to speak up for themselves, but I'll ask if somebody doesn't give me an action. It actually forces everybody to pay attention more, if you want to plan for any sort of tactics, since you have to know what each other are doing.
With the 12-player table, one of the long-term players in his first combat said, when I asked him what he was doing, "Nothing. I'm just stunned by this whole experience and gathering my wits in my first combat."
I describe the order of actions based on what I think makes sense. An orc is running 30 feet to close with you and you're firing a crossbow at him? You'll get your shot off, then he attacks. That sort of thing.
We're also shifting back to 1 minute rounds. I think it's absurd that any given combat lasts mere seconds (and the way most people describe them as one swing per roll, in less than a handful of swings). Instead of 300' per round, I've settled more or less on the old standard of 120' per round, but essentially it makes movement irrelevant. If you're in a dungeon it's much more of a question as to what's in your way than whether you can get from here to there. Outside, it makes people think more of real life tactics, like taking cover against ranged weapons. If you're a couple of hundred yards away from those archers, you won't last very long if you just charge straight in.
I originally thought of trying to split it up so that move actions were separate from attack actions, etc. That is, the question would be, "What do you do this segment?" and iterate them from there. But the reality is, what I think really matters in RPG combat are the things that grant advantage/disadvantage. To try an model a real combat, with split-second decisions, etc. is too difficult.
The other approach, which I think flanking is a good example, makes it too board-game like. "My ally is there, against that opponent, so I'm moving my 25 feet to here so I'm on the other side and get an advantage." And the target doesn't see that coming and move? As long as there's someplace to go, he'd do everything he can to keep both of you in front of him (the easiest of which is circling one opponent while keeping the other one beyond him). That's why to surround somebody, you really need 3 people.
The other thing that's cool about it is there is no shift into combat mode. Normally, the call of "roll initiative" immediately shifts everybody into combat mode. It may be that no action has taken place (which is one of the reasons I don't like it), but the DM knows they are about to be attacked. Instead, they get clues, they might be suspicious of the guy that's hanging back, maybe make an Insight check (I do these all the time as passive checks anyway). The guy lunges to attack, how to you react?
In that scenario, I'd roll for surprise. Depending on the actions and what the players said before, they may have advantage on the check. Of course, if they thought it was a different person that was going to attack they might have disadvantage. Once the surprise portion is over (mine is: you have advantage on your initiative check on that turn if needed; you have advantage on the attack; and the target cannot use a reaction against your first turn). After that, it's just combat, and you might need initiative, you might not. Note that surprise is individually, that is, some people can be surprised, and others not. On both sides.
It's very difficult to model that scene with RAW.