The biggest problem is you have to set up every device independently with accounts, senders, signatures, and have one device online all the time to apply rules. It's a lot of work to keep 3 mail clients all set up the same. Especially when clients all have their own bugs (ie; thunderbird has a CalDAV bug that makes it forget your password).
We need essentially self hosted gmail, where you have a web client for PC use and an app for mobile, mail is processed server-side, and settings are all automatically the same across clients.
Do you have a client you like for desktop? Thunderbird is not great, it's slow and buggy.