I use Thunderbird on several machines, and I use broadly the default config (no fancy business). I also have the same email accounts set up on my Android phone (Gmail ones on the native Gmail client app, an Outlook one on the Outlook app). When accessing my email on a machine which doesn’t have Thunderbird set up for me (such as my corporate laptop), I just use the webmail interfaces.
And it all works…fine. why wouldn’t it? Thunderbird and the Android apps just send their service calls off via IMAP and it all sorts itself out without any fuss from me. All the data lives off in the cloud anyway; it’s just a different way to interact with it other than the web interface.
I just happen to like having all my email accounts in one combined place, running in the background and throwing system notifications.