I'm not sure what your question is (you think the other line is better or worse?)
The point is black starts the puzzle up a piece, so by making an even trade you stay ahead. Stuff you're saying about the a file doesn't matter since black is up a whole piece.
https://www.chess.com/puzzles/problem/1110500
https://www.chess.com/analysis?fen=r2q1bnr/pp1n1kpp/2pp4/8/3PP1b1/5N2/PP3PPP/RNBQK2R%20w%20KQ%20-%200%208&flip=true
You are Black. Solution is: Ng5+, Qxg5, Qb3+, Be6, Bxg5, Bxe6. The puzzle stops at Bxe6 but as white, axb3 seems the natural next move, which opens the a file for the rook.
Now back to the beginning...I was expecting and looking to play this line: Ng5+, Qxg5, Bxg5, Bxd1, Kxd1. There is no difference in material (black still loses the queen and bishop on the move after the puzzle, and white still loses the knight and queen), but now the a file rook is still pinned, and the king cannot castle if it takes on d1. If king doesn't take on d1, then you're up a bishop.