an ordinary chessplayer wrote
on 02/15/22 at 03:02:29:
1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. Bxc6 dxc6 6. d3
5.Bxc6 is not bad. Compared to 5.O-O Be7 6.Bxc6 white is not committed to O-O, but on the other hand black's bishop is not so good on e7 either. I see that I drew against an IM with 6...Bd6 and won against an NM with 6...Nd7. Both moves are fine. If black allows Nb1-d2-c4xd6 it can become drawish, which is why I varied against the NM. They both castled kingside.
If you like this for white, it also provides an answer to 3...Nf6 (4.Bxc6!?). Here is an old game of mine:
Amateur (1900) vs an ordinary chessplayer (2300), club swiss rd.1, 2001.07.15
1.e4 e5 2.Nf3 Nc6 3.Bb5 Nf6 4.Bxc6 dxc6 5.Nc3 Nd7 6.d4 exd4 7.Qxd4 f6 8.O-O Bd6 9.Qc4 Ne5 10.Nxe5 Bxe5 11.f4 Qd4+ 12.Qxd4 Bxd4+ 13.Kh1 Bxc3 14.bxc3 Be6 15.Be3 O-O-O 16.a3 Rhe8 17.Bg1 Bc4 18.Rfe1 Rd2 19.Rac1 b6 20.h4 c5 21.Bh2 Bb5 22.e5 fxe5 23.fxe5 Bc6 24.Rg1 Re2 25.Rcf1 Kd7 26.Rc1 Ke6 27.Bf4 Rf8 28.Bg3 Rf3 29.Kh2 Rxc3 30.Bf4 Kf5 31.Rcf1 Kg4 0-1
White also has 5. d3 5. O-O and 5. Nxe5 but theoretically all 5th moves look drawish in this line, not that there's anything wrong with that.
Against the Berlin, I might wait to exchange on c6 until theory calls for it or at least until Black played a6. Here's an attempt to have a similar structure as the original model game with d3 and Nbd2.
3. Bb5 Nf6 4. d3 Bc5 5. Nbd2 O-O 6. Bxc6 dxc6 7. Nc4 with 82 games listed on 365chess +29 -12 =41
4. d3 d6 5. Nbd2 Be7 6. Nf1 a6 7. Bxc6 bxc6 8. Bd2 with similar ideas as the original game in post #1.
4. d3 d6 5. c3 a6 6. Bxc6 bxc6 7. Nbd2 with chances for both sides.
4. d3 d6 5. c3 Be7 6. Nbd2 O-O 7. Nf1 a6 8. Bxc6 bxc6 9. Ng3 with chances for both sides.
Needless to say, even though the goal may be to exchange with Bxc6 it may not always be correct. Theoretically, White scores better by not exchanging on c6 in some of these lines.
However, in the original line where 3. Bb5 a6 4.Ba4 Nf6 (Black's most played move) the continuation 5. Bxc6 looks pretty good if followed up properly.