A flaw or a blindspot came up when I did team initiative.
What do you do if PCs want to fight against each other or fight over getting some item, etc. (Like say, one of them is trying to steal the MacGuffin for their employer and another PC wants to stop them, when this happens in the middle of combat.) If they're all going at the same time...? They're not really on the same "side" anymore... but then there's the monsters too.
There's also people not coordinating and just stepping over each other, like someone wanting to fireball and everyone else just running into the melee every time and ruining it...
As with all such issues in that arise using "sides" or any other initiative--adjudicate it according to the situation. Nothing is stopping you from dropping into a more precise initiative system for a round or two--for the whole encounter or for just that "side" or for just those two or three contending players. Or for that matter, nothing is stopping you from requiring a one-off Dex or Initiative or whatever roll that fits that situation. Maybe at the moment, Int or Wis makes a better proxy for who gets to go first. Maybe the person standing right next to the thing gets a bonus.
Really, the problem with cyclic initiative is that it got built because:
A. There are some "order" situations that come up that beginning GMs don't know how to handle, including the give and take of the players intent.
B. Cyclic initiative provides "an answer" for a lot of them.
C. So to get those advantages that we only need part of the time, cyclic initiative layers on additional handling that often isn't needed.
Worse, as with any overly complicated system in RPGs, it can be counter-productive. It's relatively slow. Using it vastly slows GM development of their own skills handling those issues because they don't handle them.
Whereas, if you think of cyclic initiative as a very precise tool to handle a particular set of issues--and only use it when those arise--then it is just another tool in your toolbox. Or better, think about what it does, and use that analysis to inform your rulings.
Most people, with informed reason, don't see any need to use a precise initiative system out of combat, at least most of the time. Going around the table or just letting the players pick what they say works fine--and that's about as efficient as you can get. If your combats are clear enough, that may work fine for those too. The more players and monsters, the more that is going on in the game, the more complication for the GM and players to process--the more likely you are to need a "system" to manage that. If only to tell Joe to wait a second while you finish up with what Mary is saying. Play around with that for awhile, and you'll probably develop some modest "initiative" system that works well for your group, because it is the way you keep adjudicating anyway.