Isn't local storage pretty good for compliance? It doesn't automatically send the data to your server like a cookie would, so as long as you don't send it, you haven't collected it.

I have a hard time imagining a GDPR complaint in which storing ones own data on their computer would constitute a privacy violation

As long as you do not transmit the data ever, using localStorage isn't a GDPR concern other than that you should mention it (and the in-browser-only use, with data never leaving the users' systems) on your privacy policy page for the sake of transparency and to avoid users asking about it.