[phish]

Your example is really horrible. Yes, having the monster hand is certainly much better than having the initiative. No reasonable person is folding AA against any but the most timid well-known player even for a turn checkraise.

Where initiative matters is when neither has much of a hand. There is a subtle psychological mechanism at work in poker that is partly ego-driven: the person who takes the initiative expects the other player to defer to him. So that when the other player donk bets instead, the first player is 'outraged' and will more likely raise to reclaim his initiative, even when he has nothing. Now the other player also knows this, and unless he's willing to spray some bets, will now be more likely to defer, since he recognizes that his odds of stealing the pot cheaply is much reduced. Now this psychological game is not absolute of course, and hence there is much donk betting with nothing. But in general, it is a subtle understanding that both parties do adhere to to some extent.
Now one could take advantage of this 'understanding'. For example, let's say you flop a strong hand. Rather than checkraise the flop or try for a checkraise on the turn, you may just want to call the flop, and then hesitate and bet the turn. You will often get raised by someone without much of a hand.
But regardless, 'having the initiative' as you call it is a real phenomenon and has a +EV overall.