OP can you post clear pictures of the controller board, both sides so we can see what you've got.
I wouldn't rule out a mod quite yet.
Changing over to T12 controller may be a poor idea if the power supply has no 24VDC with enough current to drive a heater and motor. The old heater is likely 24VAC with triac control, with a DC motor?
There are T12 controllers with extra MOSFETs to switch a DC desoldering pump, as well as control soldering temp. and they have switch inputs.