iTunes account login doesn't work - I've tried it all...

Dear Community

I already tried a lot. Any additional inputs would be very much appreciated!

Problem:

All of a sudden I can't get a reliable connection to my iCloud Music Library (iTunes account) anymore.

No changes made to network settings, no Firewall activated, no little snitch in use. Internet connection via Wifi Airport extreme.

Attempts to login fail 4 out 5 times with constantly changing following symptoms:

iTunes telling me Apple server can't be reached.

Login dialog window disappears but I stay logged out. After several tries I manage to successfully log in.

"Connecting to iTunes" dialog window stays forever (e.g. 10hours over night) without ever reporting an error message

Checkbox for activiating iCloud Music library toggles itself off automatically whenever I toggle it on -> after login via "activate iTunes match" it toggles itself on automatically

Once logged in non of the iCloud Music library related actions are working properly.

Making available (showing my list of songs) of iCloud Music content fails with 99% certainty. Grey spinning wheel keeps turning without any progress ever happening.

In very rare cases suddenly my songs list is appearing nonetheless. But I am not able to download any of the content to my MacBook. The download gets stuck immediately and won't proceed anymore. (Grey spinning wheel again.)

In fact I cannot get content out of the Library into my iTunes nor can I upload new content into the Cloud.

This is only affecting iTunes for the Mac. On my 2 iPads an 1 iPhone iCloud Music Library is working fine.

Also Mac Appstore (on the Mac) is working fine although I use the same User ID as for iTunes.

For iCloud Drive I am using a different account. Everything works fine.

Interesting: even trying to log in to my itunes account via safari browser takes forever. Several minutes until the connection is established.

IMPORTANT DETAIL: As soon as I connect my MacBook to internet via the iPhones personal Hotspot... everything works fine. The list of my songs is synced from the Cloud and I can download/upload content. So it seems to be all about the network. Still I can't the problem solved.

Things I've tried and checked without the slightest success:

Log in/out of iTunes account on all devices

Deleting activated/authorized devices from the list in my account overview. Activating them again

Cookies in Safari are activated

Reset settings and alarm messages in iTunes

Reinstalled iTunes from Apple download page

Reinstalled MacOS Sierra on blank Harddrive. Tried to connect to iTunes account from a blank, freshly installed MacOS -> same issues.

Reimported programs and documents from timemachine without importing settings

Deleted network profile on the mac. Created a new network profile.

Factory reset Airport Extreme. Configured a new Wireless Network.

Checked Host settings as described in Erweiterte Schritte für die Behebung von Problemen mit der iTunes Store-Verbindung - Apple Support

Deleted /Users/Shared/SC Info

and so on and so on...

Except from my connection to the iTunes account my internet (speed, reliability) is working perfectly fine from my Macbook as well as from all other devices in my household.