Isn't White giving up too much, i.e. one tempo in the opening and the bishop pair, for too little, i.e. doubled pawns?
For example, after
[FEN ""] 1. e4 c5 2. Nf3 Nc6 3. Bb5 g6
- Bc6 is the second option in Lichess Masters database.
Some things that are probably part of the answer, but probably not complete and concrete enough:
1...a6 won't come. There is no threat to a pawn on e5, the knight isn't pinned, the only point of 3.Bb5 is to exchange it on c6. So black doesn't waste a tempo on forcing white to do what he was already going to do.
On the other hand, black has a choice to make, between ...bxc6 and ...dxc6. By taking earlier, white can see which way black goes and set up his pieces accordingly. If, in contrast, he takes later, black can first see where white puts his pieces and then choose which way to take. So since it's going to happen anyway, better do it immediately and not give out more information just yet.
A few things:
1) White's actually not wasting any tempi. He has to move his bishop out anyway (in order to castle). Then, once on b5, taking on c6 doesn't waste a tempo since Black has to spend a tempo recapturing.
2) The doubled c-pawns are more of a big deal than they'd be in, say, the Ruy Lopez Exchange variation. Since Black has a pawn on c5 instead of c7, the weakness of the doubled pawns is more prominent.
3) You're right that White's giving up the bishop pair. This is the main reason the Bb5 systems are not the main line, and instead playing an early d4 is. It's really a matter of taste: if you believe giving your opponent doubled pawns is worth sacrificing the bishop pair, you should play the Bb5 system. Many people don't believe this.
4) As for not waiting for ...a6, White has to take action before Black changes the situation. For example, he may play ...Nd4 after developing more, or he may protect the c6-knight with a piece (e.g., ...Qc7), thus preventing White from giving Black doubled pawns.
A lot of it is simply to create an imbalance. Nakamura has mentioned this before about a different opening B-for-N trade, and sometimes it is done without even getting doubled pawns in return, like an early Bg4 in the Slav then putting the pawns on c6 d5 and e6 to retain some control of the light squares.
Even in the Caro-Kan Two-Knights variation you see the same trade: 1.e4 c6 2.Nc3 d5 3.Nf3 Bg4 and the B is often traded off after 4.h3 Bf3 5.Qf3 (In the 1960 World Championship match against Botvinnik, Tal even played 5.gf to imbalance the position even more. )
The main reason is that White does not want to face
Nd4in many lines. They want to make sure they will "hurt" Black's pawn structure with
Compare this line with others where the Black knight is actually pinned, like
3...d6 for instance. There, White has no reason to hurry and trade so quickly