Why you would you run $SYNC for this scenario?
How to Reset different SAP buffers - Basis Corner - SCN Wiki
Resetting buffers will not fix this issue, is typically restricted to Basis team only and is a last resort to avoid system restart.
I'd hope in a Production environment that S_ADMI_FCD object has been adequately restricted to prevent anyone running this command.
Please do not run this command
Based on re-reading the issue it almost sounds like a PFCG profile generation issue or PFUD user compare problem. Checking SU56 user buffer (not the same as $SYNC) would show if the user is actually getting the P_ORGIN values or similar. IF SU53 is showing as failed for the user but the user has the access then it sounds like mismatch between PFCG and SU01 or the analysis and assumptions are incorrect
Regards
Colleen