He was clearly just demonstrating the mate threat to show that white can't queen.
Other moves are more obviously losing because you win the bishop with check.
And btw it doesn't matter that black has multiple moves to choose from, since you're playing white.
Letting your king get into the mate threat seems a little off, but even if the king moves to the 3rd rank the rook will be able to take the passed pawn. Maybe 2. Kg3?
Black would capture the bishop with check and then move to the a file.
I quote myself "...even if the king moves to the third rank the rook will be able to take the passed pawn."
What I meant by Kg3 is that it seems more natural to shy away from walking into their mate trap, even though RxB+ still wins the pawn.