As @girish correctly pointed out, you don't need to enable openid to use the api. In fact, when you're using the Vikunja Frontend or Desktop app it already uses the api. The login endpoint is your way to go to get the api token.
I hope to build "classic" api tokens in the future you may know from other websites (like github) where you can create a long-lived token in the user settings and use it without a login.
@kolaente Sorry, I should have been clearer that setting the time zone was what got the app running again. I don't recall clearing any cache, despite @girish's kind advice. I'll try disabling the cache, even though I can't say I've noticed any slowdowns. It either has worked well, or not worked at all! Thanks.