I just wrapped up about 12 hours of me wondering if I should keep trying to figure out this problem or if I should just quit and start filling out job applications.
It turns out that if you make a change to a SharePoint list that a Model PowerApp links to then it is possible your application can get in such a state that you cannot even edit or save any changes to it. Completely unusable. Completely broken.
Now imagine that somebody else controls the SharePoint list you are linking to...