Nino_Bilic The back-end seems fine to me. /owa works for a regular user with mailbox. Neither /owa nor /ecp works for my admin account but it doesn't have a mailbox.
As stated Microsoft Exchange Shell fails with the errors in my previous post for both the Exchange 2016 servers after installing the patch.
Whilst that seems to imply there are issues with new-pssession, neither that nor enter-pssession throws errors when ran with -computername against either of them.
'Add-PSSnapin Microsoft.Exchange.Management.PowerShell.SnapIn' seems to load fine on both of them as well and was able to run the StopDagServerMaintenance.ps1 script with that after loading Microsoft.Exchange.Management.PowerShell.SnapIn just fine.
Have compared output of Get-PowerShellVirtualDirectory with another customer's 2016 (patched too) and didn't really notice any differences with regards to authentication settings etc.
Did note some additional output on the customer with issues and both the default website and the back-end listed powershell virtual directories on it. The Get-PowerShellVirtualDirectory command returns a 'SerializationData' line on the 2016's powershell virtual directories. This doesn't happen at 2 other customers.