That is ONLY true for certain transformers. If my transformer put out 5v, no capacitor in the world could make the RMS 90 volts.
I think you've missed my point.
If you connect a 90V transformer to a rectifier and measured the RMS output voltage, it would be 90V, minus any rectifier losses.
If you connected a large electrolytic capacitor to the output of the rectifier, the RMS output voltage will be closer to the peak voltage which in this case is 90√2 = 127V minus the rectifier loses, which will burn the motor out and is as worse than connecting it to the mains via a rectifier with no transformer.
You don't need a clean DC power supply to the motor, the motor will run quite well from a 120Hz fully rectified sinewave. It is nice to have a clean supply because it will make the motion smoother but I don't think that's important for a lathe which will have a heavy chuck which will act as a flywheel and smooth the motion.
The question is, what's the voltage rating of the controller?
If you've got the 190V version, then you can use the unfiltered 90V transformer and rectifier.
If you've got the 90V version, then you'll need to use a lower voltage transformer and stick a huge filter capacitor on the output.
60VAC is probably a bit on the low side because you have to account for voltage drop and ripple but it'll probably be fine. I'd recommend 70V but it'll probably be too much for the controller which will be rated to 100V peak.
Read my earlier posts and you will see, that that is EXACTLY what I am doing.
I just want to power the CONTROLER, not the motor.
That's easy, I'd recommend using another small 5V power supply.
If you're using a toroidal transformer, you could simply add another lower voltage winding, of thin magnet wire and cover it with insulation tape. This could be used to drive a bridge rectifier connected to a capacitor and 5V linear regulator, such as the LM7805. This will save the cost/space of another transformer but you'll have to figure out the volts/turns and go to the trouble of winding another coil.
If you're using a traditional laminated transformer, it probably won't be easy to add another winding so you could use a small 9V 3VA transformer to drive the rectifier, capacitor and linear regulator.