Laurens hi,
the error happens after the initial feed discovery but before the session starts. this suggests the new client is making an additional api call to a service that the old client did not use, and that service is not reachable over your private network.
first, look at the network traffic from the endpoint device. use the browser's developer tools, f12, and go to the network tab. reproduce the error and look for any http call that fails with a 500 status code. the url of that failed request will tell you which specific service is unreachable. it is likely a new metadata or diagnostics service that the new client depends on.
microsoft is constantly adding new features to avd, and sometimes the required backend services for a new client are not all covered by the original private endpoint configuration for the workspace. you might need to open a support ticket with microsoft to ask for the complete list of fully qualified domain names that the new web client requires for private link scenarios. their documentation might not have caught up yet.
as a temporary workaround, while you investigate, you could create a specific dns rule. if you can identify the failing endpoint from the browser logs, you can create a conditional forwarder in your internal dns to resolve that specific hostname to its private ip address, if it has one.
this kind of issue is not unique to azure. any time a platform rolls out a new client, it can have new dependencies that break in locked down environments. it is always a game of catch up for network security.
so, to summarize. use the browser's f12 developer tools to find the exact url that is failing with a 500 error. that is your smoking gun. then, you can either adjust your private endpoint to include it or provide that information to microsoft support so they can fix their documentation or service.
rgds,
Alex