Just an update:
I raised a ticket for the Avast Support, but their first workaround suggestion doesn’t help.
They suggested to put the Remote Desktop Services to ‘Automatic’ and make sure it is ‘started’,
and (again) do a clean uninstall and complete re-install, but this doesn’t help.

While looking in the Windows Eventlog, I can see following entries:
at 17:43:59 event 7031 - Service control manager : “Avast Antivirus-service stopped unexpectedly. This occurred 1 time. Next repair action to be executed in 5000ms: restart service”

=> apparantly the repair action (restarting the service) worked

at 18:12:52 event 7031 - Service control manager : “Avast Antivirus-service stopped unexpectedly. This occurred 2 times. Next repair action to be executed in 5000ms: restart service”

=> apparantly the repair action (restarting the service) worked

at 21:01:17 event 7031 - Service control manager : “Avast Antivirus-service stopped unexpectedly. This occurred 3 times. Next repair action to be executed in 5000ms: executing the configured repair program”

=> apparantly this repair action didn’t solve it, since now Avast is stuck. After this event, another event is reported: event 7032 - Service control manager : " executing the configured repair program failed with following reason: no repair program is configured for this service"

Note that the text between quotes is my own translation from Dutch to English (so in English it won’t be literally like that)