Apple - Contacts and Phone on iPhone think I'm in a different country
When I was writing this question and was in Language & Region, I tried setting the iPhone Language even though I didn't expect it to do anything. It was already set to English but I went ahead and selected "English (US)". It went to a black screen that said "Setting Language" (or something similar) for a while and when it was done all the phone numbers are displaying correctly. So I guess it's possible the language wasn't set to "English (US)" but I can't imagine why. Anyway, since I already went through the effort of editing the screenshots and writing the question I figured I'd leave this here in case it's helpful to someone.