I am Canadian, no worry about getting a big medical bill. Of course I would go to the er to get it checked out and cleaned.
This. I'd probably pull it out myself, without twisting it, just straight pull, that way if there are any metal shavings I should hopefully pull those out at same time as they'll get pulled in by the thread. will hurt more as I might pull flesh with it too though. I'd maybe use a hard drive magnet to try to get any shavings that did stay inside, then run it under cold water to try to clean it up and slow down bleeding then dry and bandage it, and head to ER so they can check it better.
The thing with this type of injury is it usually does not hurt right away, it's the day after that it hurts badly. May as well treat it asap while it's not hurting as much. ER might also freeze it so they can go inside it with a scalpal or other tool to inspect it better, so there's that.
I refuse to use a power tool near something that someone else is holding and vise versa. if I am holding my own stock that I'm drilling or cutting into, if I slip up, I'll be able to react faster to get my hand out of the way than if someone else is doing it. I try to avoid situations like that though, sometimes you have awkward cuts to make, but I do my best to keep the procedure as safe as possible.