I'm using Thunderbird with POP3 accounts, download on my laptop then remove from the server, daily backups. My phone has K9 (the old one with the UI I like) and I never remove messages from the phone. When I send mail from the phone I bcc myself so I can see the message from my laptop later.
Why? Because I don't like to leave my mail forever on somebody's else server. They can lock me out at any time (it will probably never happen) and my mail is mine.
Would I recommend this to anybody? Of course not.
Is it a problem not to have access to all of my mail when I don't have my laptop with me? It never happened to be a problem and it's always less likely to be a problem because of all the messages that are exchanged outside email and on mobile first platforms, even for work.