There's got to be some way of deleting mails from a POP3 account.
Yes. You set up your email client to delete emails after they are downloaded. This is the typical setting for a desktop's mail client. The iPad, by default, leaves emails on the server after downloading, but this can be changed in the account's settings.
There is no server side syncing with POP3. Once downloaded the email is local on whatever device downloaded it.
If you have only one device/computer accessing a POP3 account (how it was designed to work) then you should have your client set to delete emails after downloading them. If you have more than one device/computer it is common to set one (main) email client to delete after downloading, and all other's to leave them on the server. This makes sure that the inbox does get cleaned out, but that other devices get an opportunity to see the email; provided you don't use the main client first.
Others will set all devices to delete on download, and simply deal with the fact that emails will be scattered across several devices.
A few people will tell all clients not to delete.
This seems to work for a while, but has two problems. One, POP3 servers typically have storage limits. Eventually you'll exceed this and your email will stop working. Two, if an email account gets removed and added again, it will often see every email on the server as new, and needing download. This is the usual cause of the 'My iPad just downloaded over 3000 old emails, how do I delete them all' post.
While there are a few exceptions where POP3 is desirable, in general, anyone who uses more than one computer or mobile device to access mail will find IMAP or Exchange account much easier to manage. Though, even there they must make a clear distinction between folders on the server and local folders if they want to be sure to keep everything perfectly in sync.
There is one other method of deleting emails from a POP3 account. If the email provider has a webmail site, then that site will typically show your inbox as it is. I have no idea how webmail does it's thing, but it is not POP3.