Finally got through to the web resolution team over the phone just now; my CM at least seemed to acknowledge it's a known and widespread issue. We sail on Monday so we were able to check-in without issue last month. She spoke with a supervisor and they're putting in a ticket for me, but basically the only advice at this point is "keep trying."
Wanted to pop in here and say she DID tell me they've been assured the app works fine once onboard and on the ship's WiFi