[VanVeen]

The short answer to your question is simply, "no". There is no inherent advantage to having the initiative at any form of poker. "Initiative" is shorthand for exactly what you've suggested it is: having a stronger hand range and making the perceived cost of rebluffing with nothing hands exorbitant. The information leaked on the previous round(s) of betting by all the players involved makes it so. "Initiative" would have no meaning in a game between players playing optimally from a game theoretic perspective.

But that doesn't matter and you already knew the answer. What's important is: Given how the overwhelming majority of players play poker, 'initiative' is something worth seizing and the language used on this forum is entirely appropriate.

You seem more interested in poker from a theoretical point of view while nearly everyone else here is interested in taking money from donkeys. For instance, there is no reason whatsoever to 'balance your hand range' by randomizing 3bet turn bluffs or flop donk bets when even expert players are making easily identified errors that you can exploit with simpler strategies. Even the way in which they adjust is exploitable. You have to keep in mind that they're working with very limited information when it comes to your play, and as a consequence they're forced to overadjust in many cases (because they correlate your play w/other players they've played against who employed straightforward, exploitable strategies). You don't need a non-exploitable hand range: you just need one that exploits your opponents' hand range while hopefully triggering desirable adaptations (or avoiding undesirable adaptations).