And a good example of why you won't see any 737's (and probably 757, 767 either) using reverse thrust for pushback anymore is the lessons learned from Air Florida Fight 90 some years ago.

Air Florida Flight 90 initially attempted to pushback from the gate. The tug being used was too small with the ice and snow on the ramp, and the aircraft could not be moved. A powerback was then attempted. Although the
engines were run with the reversers deployed for someting like 30-90 seconds, the powerback was also unsucessful. A larger tug was obtained, and the aircraft was pushed back.

The board determined that ice and snow blown forward during the attempted powerback was most likely a very significant factor in the subsequent P2 probe becoming iced over, and the higher than actual EPR indications.

The snow, ice, and debris blown around from the redirected thrust was a factor in clogging/obstructing the P2 probe. On takeoff, this led the airspeed indicator in the cockpit to show a higher indicated airspeed than the 737 was actually traveling, a problem which continued right
into the stall/impact.