I agree, the voltage shouldn't be critical, nor should the power supply amp rating, as long as it isn't ridiculously low. 12v, 1 amp, should be plenty. The big capacitor is the thing that fires the turnout coils, and you have time to charge that up.
Maletrain:
As for how the transistor circuit compares to your original drawing, it really doesn't make any difference at all other than speeding up the recovery time. I could wire two DPDT momentary contact buttons into mine to allow it to fire the coil in both directions (a detail I didn't put in that schematic). In your case, you are using a toggle and pushbutton.
I still think that if it were me, I would still prefer Stilwell's circuit because he exploits that trick of the inline LEDs to indicate turnout position, and only a single control is needed to activate the turnout (either pushbuttons or a single toggle, and with the toggle, you don't need LEDs because it indicates position). That, to me, is the best user interface and visual feedback. The low-level ripple going through the turnout coil would be no drawback at all. It's not going to heat up or burn anything, nor will it make any noise.