How to monitor a file share witness for accessibility
The file share witness does not need to be accessible by the SQL Server service account. The FSW is used by the Windows Server Failover Clustering cluster to achieve a quorum and avoid split-brain scenarios. The cluster machine account is the AD principal that needs access to the FSW and if it did not have access you wouldn't be able to set it up as a witness in your cluster.

From an Availability Group perspective, it does not care about access to the FSW. The cluster cares about this and when it fails (and the quorum is lost) it signals the AG to shut down to avoid split-brain. The AG only cares about the SQL Server specific health checks (latency, connectivity to replicas etc) and the signals received from the WSFC cluster regarding cluster health.

If the cluster CNO loses rights to the FSW share then that resource would go offline in the cluster. This won't necessarily trigger any action in the cluster but should log alerts and events in the Windows Event Log for which you can monitor.