Yes, a carbon brush is the usual solution, but it is much more involved than just clipping an alligator clip to some metal parts. It will also eventually wear out and might get caught in something while in use

If this was my machine I’d probably just use alligator clips and a jumper wire in the future. Whole house RCDs typically trip at about 30mA, so you are still well clear of that happening

The limits we look for in these cases (in germany) are:

1. less than 50V AC and
2. less than 0.5mA

(This is for touchable metal parts, that are not connected to earth)

So yeah, I’d try to avoid touching the motor spindle, but with that low of a current it shouldn’t actually be a problem as any current flow at all will make the voltage drop significantly.

You can try grounding as many of the metal parts as possible, but the motor shaft will likely be difficult