Update: For the past month (4 stays) my Hyatt Dual Accrual miles (HDA) and corresponding LP have posted to AA as expected on the first Tuesday after checkout unless checkout on Sunday or Monday in which case the second Tuesday. Still missing ~5,500 HDA and LP from stays between Feb 1 and Apr 8.
The Hyatt Points-to-Miles conversion (post #45 above) posted to AA a week after the Hyatt activity date, no loyalty points.
The last American Advantage Executive Liaison Team agent I spoke with has actually called me twice since post #45, exactly 7 days between the 2 calls. First time left a voicemail, second time we spoke in person, agent said could see the Hyatt xfer, said would still work on trying to credit missing miles and LP but wasn’t certain they could post LP manually.
Bottom line: Right now the automatic system (for my accounts) seems to be working again. I’ve spent several hours dealing with it. I’ve received approximately 3 times the number of missing miles (as Customer Service miles) and zero LP. Both MHC and American Advantage Executive Liaison Team have been diligent and responsive and (pleasantly) surprised me with follow-up emails and phone calls.