Why do players play this move (bishop to b6) in bullet chess? I've often seen (on lichess, I play only bullet, at ratings 1800ish) black play the f8 bishop to c5, and then if white challenges it by playing their c1 bishop to e3, black retreats their bishop from c5 to b6, and then if white captures, black recaptures with the a7 pawn. 

rnbqkbnr/pppp1ppp/8/8/8/8/PPP1PPPP/RNBQKBNR b KQkq - 0 1

1. Bc5  Be3 2. Bb6 Bxb6 3. axb6


Why is this a good move? I've see it CONSTANTLY in a variety of openings, and I myself never play this way. 

It seems a bad move for black to me because 

a) you lose a tempo.

b) you double your pawns.

c) yes, you do get an open file for your rook, but ... so what, quite frankly? It's a bullet game. You are probably not going to have the time to think of some ingenious rook lift in the middle of the game. It just seems too risky to utilize that a-flank with the rook in a bullet game, at least for the players at my level. Indeed, I can confirm that I can't recall a single game where black utilized the strategy above and actually got something worthwhile out of it. The rook never made an impact on the a-file in a single bullet game that I've played where black played above strategy.

So why do players do this?
Actually, ..Bb6 is a sound move not just in Blitz. The Italian and Scotch game are filled with it. There are plenty of reasons why you would make it:


After 1.e4 e5 2.Nf3 Nc6 3.d4 exd4 4.Nxd4 Bc5 5.Be3 White is threatening to win a piece, so Black can retreat to b6 and prevent his bishop from hanging.
Let's assume a central formation like e4+d3 vs e5+d6, with a Black bishop on c5 and a White bishop on e3. It is good for Black to retreat to b6 (even if ...Bxe3 or even just leaving it on c5 could sometimes be considered) in order to trade them on that square and open the "a" file. In your post you give a lot of importance to the fact that this doubles pawns and very little to the opening of the file. But I would say that, specially in Blitz, the file is way more important! White will have to worry about the a8 rook for the entire game.
Finally, Black could have retreated to b6 (or a7) to prevent a trade and keep a powerful bishop on the a7-g1 diagonal.