Still working for me in the US, no VPN, legal account, Google DNS.
Cache/temp cleared regularly with each update. Might be some change to Prime Music rolling out that has not hit everyone yet.
Post your log file showing the errors generated when trying to login, after first clearing your cache/temp to get clean log file(s).
One trick I also do is once logged in, actually play some music by an artist before downloading, but if you can’t login with that pesky message, then you’re stuck waiting for update.