My question was: what would happen if black did not move his bishop to b4, but rather took the pawn at f4. So far, the answers I keep getting begin with b4
which in all cases stop the threat of 2... b1=Q because now 3. Rxb1 is available
The move Bb4 has two functions: clearing the way for a possible discovered check in the second rank, and shielding the promotion square from white rook. On the other hand, the move Bxf4 only clears the way for the discovered check, but doesn't protect b1, so promotion is no longer possible.
Black could always move the rook to a1 before promoting the pawn. It forces a trade, but still leaves black with an advantage.
This is too slow. After 1... Bxf4 2. Ng2/Kh1/Kg1 Ra1, white can simply take the pawn for free with 3. Rxb2, and black loses all the advantage.
If white moves his knight to g2, check with the bishop at g3 also gets him another pawn and opens up the king’s defenses a little.
Trading a bishop for a pawn (one pawn that wasn't passed and only in the third rank) is not a very good deal, and black makes no progress towards promoting. Also, in endgames, king activity tends to be more important that king safety.
The bishop is in danger and unsupported no matter where you move it.
What I meant with giving check with the rook was:
1… Bxf4
2. Kh1/Kg1 Ra1+
That puts white in check with the rook, forcing the king to move, followed by b1Q. If then white takes the new queen at b1, she is protected by the rook at a1. Again, not the most elegant solution but at least black is not giving his promoted pawn away for nothing