I found that the cname record seemed to be the problem. Here someone had the same issue. After changing now to a wildcard A record pointing to the tailscale ip, it all magically works. Thanks for all the answers!

Yes, correct. So far I was able to access the services via <tailscale ip of the host>:<port> on any device in my tailnet.

I don’t have a problem to always be on the tailnet with my client devices, but it does not work even for this case.

Yes, similar to the video guide, I set up a wildcard CNAME record to point to <machine>.<tailnet>.net.