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Just solved the problem, somewhat at least. While private networking still didn’t show up as enabled under settings, when I went to Console Access, the private IP and related info was all listed, and I was able to use that to finish setting up private networking. I’ve tested it and can ping the private IP from another droplet in the same region, so it definitely works. No clue still why it’s not showing up in settings though.
So it looks like private networking is actually turned on, but it just doesn’t seem to update in the settings page. The workaround I found is to go to the console access page, and at the bottom, for me at least, the private IP was listed, and it seems to be correct.