Call done, the support representative has been very kind and acknowledged that this issue is likely to be server side.
She also noticed that even yesterday she saw some clients on High Sierra with this issue but she didn't think it was a widespread issue until my call.
She told me that she will raise the issue internally and suggested me to fill in a feedback report through apple.com/feedback, too, in order to prioritise the issue.
However, she told me to not expect a super fast resolution since macOS High Sierra is not actively supported by Apple anymore and currently only obsolete or vintage Macs cannot update to a more recent macOS version.
So, folks still on High Sierra, let's open Apple Support tickets and report this issue on Apple's feedback page!