I cut my index finger pretty bad, it must have been all the way through the skin as there was a big flap. It definitely needed stitches. I used superglue to hold it together until it stayed stuck on its own. I couldn’t use the finger for like a week because it was all bandaged up. Probably not as a good job as if I went to the doctor, but my finger didn’t fall off. HDHP turns you into a do-it-yourselfer.
My mother had a much more serious gash when she dropped a piece of glass on her foot. This was on Croatia. We went to the ER in the city, and she got stitches fairly quickly for like $20 cash. If I would have cut my finger in Croatia, I would have gone to ER to get stitches.