nltest /dclist command on some sites

Because RDP to localhost also fails after reboot, this doesn't look like a normal firewall or external-network problem. The fact that reinstalling RD Connection Broker temporarily restores service suggests the RDS deployment or one of its dependencies isn't initializing correctly after boot.

Before reinstalling anything next time, capture the state:

Get-Service TermService,Tssdis,RDMS

Get-RDServer

Get-RDSessionCollection

Then inspect the events around startup, especially:

Microsoft-Windows-TerminalServices-SessionBroker

RemoteDesktopServices-RDManagement

System

Application

Pay particular attention to the existing Event IDs 1269 and 1306 and the events immediately before them.

Also verify that the Connection Broker database dependency—Windows Internal Database or SQL, depending on your deployment—is running correctly and check whether security hardening/GPO changes are affecting RDS dependencies. Connection Broker/RDMS failures can prevent a deployment from becoming usable even though the machine itself has booted.

Repeatedly reinstalling the Broker is masking the underlying issue. Capture the failed post-reboot state and fix the service/database/certificate dependency that is preventing RDS from initializing.