Old iPad completely fails to fetch any e-mail
Since:


other boxes and
other iCloud account's aspects


are working without any problems, my suspicions came to Inbox directly.

After reading this answer and this blog post it turned out that I have mail forwarding + delete source message option enabled. All my e-mails are being forwarded to the account that I stopped checking years ago.

Turning delete source message option off resolved this particular case and e-mails started to appear / remain in my Inbox as previously.