The Ruy is one tough opening to face as Black, though I do face it without the Berlin nonsense.
As White, If someone plays 3... Nf6 against me, I play Nc3 and go to the Spanish 4 Knights. See how they like that!
More recently, many Morphy Defense (3...a6) lines with c3 & d3 have been revived.
I have played the Italian for decades and always wished that I studied the Ruy when I was a youngster. I didn't, because I didn't appreciate the intricate nature of the long-term pressure exerted on Black's King-side. When you start to understand why this works, you become a better player in general.
White develops a piece, is ready to castle, and loosens black's grip on the center all in one move. White isn't planning on winning a pawn yet, but that can easily become a real threat. For example:
As for white's bishop getting chased off with a6 and b5, black gains a lot of space on the queen side and opens up the b7 square for their bishop. In exchange, the b5 pawn can get picked at when white attacks it with a quick a4. White moved their bishop several times, but black made several pawn moves that might turn out to be a bit weakening.
So I guess the difference in your example is that d3 has prevented the Qd4 re-capture. But black waited an extra move to chase, which allowed that to happen.
It just seems like I get chased every single time, so I think "why not just stick to the Italian?"
I was thinking that the true magic must lie in how one has disrupted the a/b file - which doesn't come in to play until much later, and therefore I don't know how to take advantage. I guess I asked the question because this opening comes up in almost every beginner guide, and yet seems to be an exercise in running in and away again. Perhaps it shouldn't be in beginner level guides...
Thanks so much for the responses, and I seem to have some PMs on this too, so thanks again!
It might help to point out that pawn moves don't count as development. In other words, sure white's bishop has to "run away" but black isn't gaining any time because it's only pawn moves.
Also worth pointing out is that pawn moves are always double edged. It's not the case that a pawn is necessarily better on a further rank. They weaken the squares and diagonals they leave behind, and in general the closer they get to the other side the more vulnerable they become.
The good point is pawn moves gain territory / cramp the opponent.