Exactly that, it's terribly simple. People have multiple devices, that alone makes POP3 an annoying protocol to use.
A big assumption. POP3 works perfectly fine and is preferable in my situation.
The workload required to poll is heavy. You need to reconnect TCP, TLS and login. Once you find you have no messages, you have start over to check again. https://billpg.com/pop3-commit-refresh/
Clients and servers need to keep a short-term "message id" to long-term "unique-id" map. https://billpg.com/pop3-message-ids/
Deletes are batched and only committed at the end of a connection. Unless you're willing to close and reopen a connection right away, any messages you delete are in a not-quite-deleted limbo until you close the connection. https://billpg.com/pop3-deli/
I use POP3 only to move messages from my mail server's mailbox on a VPS to my LAN using a pull method. Polled mailbox is thus mostly empty or has very few messages at each poll. So it's a nice fit.
Having to use anything more complex would be a downgrade. More complex to setup on server and client side, with more options and more problems to take care of.
I like to use the simplest tool for the job.