and pass play back to the PCs
I'm not trying to backseat DM here, but like, when you say "pass back", what did you do? I feel like this game assumes (and doesn't quite properly explain!) a PtbA-like approach where you pick who you pass it back to a player by asking that player a question, and if you do something more like say "Balls in your court guys, what do you do?", then that is a bit of a different approach that's going to work more variably with some groups.
Also, obviously I'm not there, but to me, the "enthusiastic" player
sounds like they maybe lack some social skills/empathy (nobody's perfect!), and that if they're "getting bored" because they blew their turns and now have to wait the same exact amount of time as everyone else, that's a very fundamental problem that has nothing to do with DH - that's literally the same in D&D or any RPG where the PCs have individual initiative. That does mean your side-based initiative may be the right thing to help of course! But you're essentially working around someone who is being a bit naughty and selfish, and unless they're really obtuse, probably knows they are being at least mildly bad, and should probably meditate on that a bit!
Out of curiosity, where were the other players?
My experience is that most other players are weirdly bad at intervening on this kind of thing unless they've got DM experience themselves (or similar "small group management/fairness" experience y'know, like school/kindergarten teachers or people who lead meetings at work).
I do agree that players usually organise themselves so they all get a go in games without strict initiative etc. but when that breaks down I think a lot of players either don't quite realize it's happened or don't know what to do about it.
i am arguing against the GM taking an active hand in telling the players who should go next.
But why? PbtA expects the DM to do this, and the design of DH pretty much definitely expects the same thing, even if it lacks clarity. I haven't seen Matt Mercer's actual AoU session yet but I'll be shocked if that's not how he's running it to a certain extent. I mean, you don't go to a specific player every single time, sure, but a significant amount of the time? I think that's expected and kind of required to function correctly.
Especially as sometimes the natural flow of things is going to leave a player out more - that's not the players screwing up or being bad or w/e - it's just how it works when you don't strictly cycle through turns (including in trad RPGs).