Jeff, I've only made a few small springs but the way I approached it was the same way you are describing...by making a simple mandrel with a hold-down screw to pinch the music wire. I used my floor-standing drill press and rotated the quill by hand so I could easily control the winding process.This Old Tony has an excellent video on it:For those who already know how to make a spring, it's worth a watch just for the production value and humor.Regarding the type of wire, I use music wire.
Here is a handy mandrel calculator....https://www.daycounter.com/Calculators/Springs/If you have a lathe and want to wind by hand, make a handle for the headstock.... It tightens like the seat post on a bicycle, a taper-lock....but PLEASE make sure you UNPLUG the lathe before inserting the handle!....Bob