Bc4 vs Bb5

"In all variations of the Ruy black is struggling for a draw"
What? I take it you haven't heard of the Berlin yet? It's super solid and black right away almost equalizes already.

As for the Two Knights, who says White plays Ng5? Most white players play d3 instead, getting a normal Guicco Piano position and here it is where White gets a small but long lasting pressure. (doesn't mean Ng5 isn't playable, but White can always go for solid d3)

The Berlin is a great example of black's only ambition is a draw. Before Krammik revived it, it was considered a weak defense which black struggled for a draw. Deeper analysis has changed this evaluation, but black is still only hoping for a draw.

As I stated, when white plays the Ng5 varation, so yes the Ng5 variation is what I was talking about. These variations allow black a more active game, and I agree that the d3 and d4 lines are better. Of course, I can disregard what you wrote and claim that d4 is the best first move, because it avoids the really aggressive Sicilian.

Once again, this discussion is supposed to be about 3. Bc4 vs 3. Bb5 and the reasons one move is played instead of the other.

I thought the discussion was about the merits of 4...b5 in the Spanish Ruy Lopez.

Well, the Berlin IS the reason why Bc4 is played a lot more these days.
And black can win in the Berlin btw if white plays poorly. It's roughly even but with some small winning chances for both sides.

As for a6, b5, I'd argue that that's more often than not a weakness as black commited his Q side pawns. (and can't move them back obv)
Also as was mentioned the B on b3 is better than on c4.