It can transfer from PS4 to PS5 via a network connection. They both have to be on the same segment though.The console cant transfer with direct ethernet connection between the 2?
I think there needs to be more data. On my 4Tb external SSD I have about a 50/50 split of games I bought digitally like Ghost of Tsushima and games I bought on disc like Dying Light and both work and have received successful updates on the drive when hooked up to my PS5 - in rest mode or in the background when playing AC Valhalla, which I have been doing a lot!
The games in question that didn't work were ones that I had transferred from my PS4 to PS5 via the network transfer option. This is what makes me suspect they have an issue with their network stack, it could also account for issues that some people have encountered with rest mode.