The only reason I can think of to use a shutter in a color head, is that the bulb doesn't reach it's operating temperature + color temperature instantly, and with a shutter the exposure can be started when the lamp has reached that temperature. This may give better color consistency when the exposure times are short. Is this correct?
The single filament bulb on the 2000W head does take some observable time to warm up (even with the baseline heater current going through it) and it has quite a long orange decay after the power is cut. Back in the day, a Color lab needing to make 20 identical color prints would benefit from the perfection of this system.
In your system I agree with hooking up a generic power supply. Just make sure it has the wattage to handle the lamp you will use. Since the filters in dichroic heads don't wear out, I think it is worthwhile refurbishing color heads. Dichroic heads will outlast two-color coldlight heads.
Sticky filters can be freed up
Bulbs can be replaced or easily substitured
Bulb sockets can be replaced
Generic power supplies can be bought or made
Frayed cords can be replaced
Sealing foam rubber can be replaced
Mixing boxes can be rebuilt and relined with styrofoam
etc...