Look, I understand that you're happy with rerolling initiative, but the behavior you're describing from your player here requires knowledge of AC and hp to make those sorts of predictions. If he's using that at the table to the extent that you're describing, it's just metagaming. The solution isn't to roll initiative every round. If as soon as you put the figurine on the table your players know it's hp, AC, saving throws, etc., the solution to that isn't to roll more initiative dice. It's to prevent that metagame knowledge from being useful. The solution is to mix things up. Use Ogre stats for Human warriors, or Hobgoblin stats, or Bugbears, or whatever.
That'd be the start. Really, this is the sort of behavior that keeps lighter systems on my radar -- Fate, especially. I'm currently happy with 5E, but I would go insane with even one player that used metagame knowledge to that extent. Don't get me wrong, I love tactical games, especially board/war games that reward a certain level of system mastery and/or "4D chess". That's just not what I want from an RPG.
As you say, to each their own. In this case, it sounds like the metagaming actually is a problem at the table.
But with cyclical initiative, I could easily see too many options for tactical abuse by the players. If I always know my fighter is going before the other player's raging barbarian, you better believe I am going to try to shove the target and knock it prone! That raging barbarian then gets advantage to attacks. Oh, and did I mention he is benefiting from the Extra attack feature, has GWM, and three levels of Rogue so 2d6 Sneak attack damage? So, it makes even more sense for me to act in a utility role and let the barbarian optimize his hits. That is huge!
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But hey, many of you might not see this as abusive, so no worries.
I'd totally see it as abusive and it'd totally drive me from the game. I haven't had it as an issue for a couple of reasons:
1) I've made it extremely clear that I have no interest in running a game that's super crunchy or where PCs are cycling through ever more powerful magic items. I don't begrudge people for wanting that, I'm just not the GM for it. The end. Note: That doesn't mean I don't work with the players to make the game interesting for all, just that I have some boundaries and I'm up front about them.
2) I tend to prefer villains who are PC races. That means they're as unpredictable as PCs and have the same sorts of wildcard capabilities. Who cares if you know the math on the routine encounters when the ones that actually matter are wildcards.
3) When I do use monstrous opposition (which is common, just heavily spiced with PC races), I narrate the appearance, rather than being clear about what things are, unless the PCs would know. I don't overdo it, but I'm also more than willing to run with anything the PCs say. When running LMoP, I was describing the orcs (or was it goblins) that are all over the place, and actually called them orcs. The players kept slipping and calling them goblins (or orcs, can't remember which way, anymore). I corrected them a few times, and they even caught themselves and corrected. But, they kept doing it. After a couple sessions, I ran with it and just started calling them goblins because that's what the players had called them during a half-hour of straight planning. I've done this with other monsters and magic items (sure, it's a dragon slayer, whatever) on a number of occasions. The players know I'm doing this and are cool with it.
3) In practice, the problem with "I know I go before X" is more theoretic than realized. I was concerned, as well, mainly because I'm so strongly opposed to the aggressively crunchy game style. Even for the tactically minded and borderline char-ops players I have, I don't think I've yet seen any time it's been an issue. Whether that's because of the above or not, I couldn't say. It just hasn't been an issue.