Is Bxd5 really that bad here?


Black has a bad bishop on c8. Bxd5 rxd6 activate the black bishop. Re1 is really better for white.
Good point! I did not see that. Thanks!
Bxd5 gives up the bishop's pair. Bc4 is white's good bishop, not on the color of its own pawns, while Bc1 is white's bad bishop, not on the color of its own pawns.
Re1 is a good move, as it activates the rook.
There’s no real immediate need for trading, and in the long term normally bishops could be better in open positions, and immediately the development of the bad bishop of black becomes very easy and it stops being a bad bishop. In this position I would be mainly looking at mainly Re1, Qe2 or Qc2, probably failing to play Re1.

See it like that: Black has a "good" knight at d5 which does nothing special, he lacks space, as well as a decent way to put the Bc8 in the game. Also, his kingside (especially h7) is vulnerable as only the king gives protection, so the plan would be: Avoid exchanges which allow Black to breathe, and organize a kingside attack.
The most natural move is 10.Bd3, but it does allow Black to play ...f5(f6) with a hint of counterplay. So, one has to pick between 10.Ne4 and 10.Re1, and I do not think one of them is better than the other.