If your "in one place" reader is still open and downloading messages then there will be no messages to view in the web interface when you have to.

There will, because my client doesn't delete the messages from the server when it downloads them.

POP3 is more for reading and acting on your email in one place (taking notes, plan actions, discard and delete,…). No need to consume them on other devices as you’ve already extracted the important bits.

I use imap on my mobile device, but that’s mostly for recent emails until I get to my computer. Then it’s downloaded and deleted from the server.