After 2. Kc1 f1=Q 3. Qd1+, white's position is hopeless after queens are traded. Surely e.g. 2. Kb3 gives white a better chance, keeping queens on the board.
2. Kb3 may lose more slowly, but I don't think it gives White any practical chances, especially as there is no playable check after 2. ...f1=Q. The 2. Kc1 line is the critical one to demonstrate why 1. ...Re2+ wins.