That’s the issue 185400 describes, the cookie context between Safari and a PWA is no longer shared.
Sadly Apple doesn’t make the status of the stuff they’re tracking on Radar public, so until they issue a fix there’s really not much that can be done, short of the app you’re pursuing.
They stopped signing iOS 11.X a while back, and even iOS versions prior to 12.1 are no longer being signed, so there isn’t even a way to roll back to a “working” version of Safari.