Here's what I finally managed to carve into the PCB material:

Unfortunately I carved away the copper at the zero coordinate, and thus I couldn't measure the drill bit, so I just drilled the holes with the carving bit to see what it looks like.

I'm still using the broken bit BTW, so IDK, but an unbroken one might cut better.

Here it is after some sanding. Even with this broken bit, that would be a usable result for a simple PCB that doesn't expect to use parts with less than 0.1 inch pin spacing.

I spent all day trying to get it to carve without losing connection.

I found a few people talking about the same problem on the internet. As best I can tell, somehow interference is being picked up through the USB, causing the USB controller in my PC to disconnect the thing. The message in Linux's log files even suggests electromagnetic interference may be the cause. There weren't a lot of great suggestions for how to deal with it, so I ended up wrapping the spindle motor wiring through a toroidal ferrite five times, and put the power and USB cables each through their own rectangular one about as many times. With half as many turns it still wasn't working so I'm not all that confident this actually solves the problem vs. me just getting lucky.

A note from the future: It totally fixed the problem. I did the same to my 3D printer's USB cable and now it no longer randomly disconnects either.

In any case, USB sucks, so if it happens again I think I'll see if I can make a serial connection to it and hopefully then it'll be less sensitive to interference.