I spent over an hour updating devices to get "Family" (shared) Reminders working again....and it still isn't. Then I find this, which is a bit infuriating:
...but these articles are from September. SURELY this is fixed by now....right?! All devices are on iOS 13. When I send the invite to my wife's phone we accept it but then...nothing. There was a yellow exclamation point on the "Family" reminders previously (which I erroneously assumed upgrading to iOS 13 would fix), but after the upgrade didn't fix it, I tried re-creating and re-sharing, but no luck so far.