IPad – Old iPad completely fails to fetch any e-mail

tl;dr My old iPad's Mail app does not fetch any e-mail. Inbox remains empty all the time. Everything seems configured OK. I can send e-mails from this iPad. And receive them. And I can sent e-mails to this iPad. And they're not returned with error.

I have never run into situation like that previously, so I don't even know where or how to start a potential investigation.

I have a 7-8 years old iPad with iOS 9.3.6 (13G37), model MD544FD/A. Since I am completely not using its e-mail address for communication nor Mail application, it took me several months to realize that I have completely empty Inbox all the time (other boxes like Sent are full) and I cannot receive any e-mail to it. This my only Apple device that I have, if that matters.

(all cites and text below are my own translation from Polish)

I can send e-mail to this device as normally:

• I'm sending them from my PC, through Gmail to firstname.lastname@icloud.com,
• I've sent dozens of e-mails like that past months and none of them were ever returned with error,
• Mail says "Updated: a moment ago" (after manual Inbox refresh / forced fetch), but Inbox remains empty.

Account seems to be working normally:

• I can install and update any application from App Store using this account / e-mail address,
• Settings > iCloud shows this account in second line,
• Settings > iCloud > iTunes and App Store shows this account as my Apple ID,
• Settings > iCloud > Advanced > Mail > shows this address as Email and has SMTP pointing to iCloud SMTP server.

Mail application seems to be working normally:

• since I'm not using this e-mail / account for e-mail exchange at all, Inbox, Drafts, Spam, Bin and Archive boxes are empty,
• Sent box contains about 40-60 e-mails ever sent by me directly from this device,
• I can sent any e-mail from this device and within 1-2 minutes I can receive it in my Gmail.

My Gmail shows me 20 e-mails sent to firstname.lastname@icloud.com in 2013-2020 period. None of them can be found in my Inbox, Spam or Bin. All these folders remains empty.

Directly before writing this question I've sent two e-mail to my account / @icloud.com e-mail / iPad. One was a new message, the other one was a reply to a message that I've sent to myself from this iPad just a minute ago. None of these messages were rejected / returned with error. None of these messages were received by iPad — Inbox remains empty as of writing this.

This problem is so weird to me that I don't even know where to start tracking for a source of problem or for a possible solution.

For past 25+ years, since I've been using e-mails it was always either message delivered to the account or message rejected. So far I was lucky enough to never lost a message, i.e. like with this iPad — message neither received nor rejected and returned to sender.