Been using mailbox.org for a couple of years and I’m quite happy with it. Don’t know what “needs client app” is supposed to mean though. If you don’t want to use a mail client you can use the web interface. I’ve used maildroid on Android with it and when it was discontinued switched to Thunderbird, both work fine with mailbox.org.

There’s one tiny minor annoyance, they somehow automatically create a new “archive” folder for every year that the web interface uses when you hit the archive button but in your mail client you gotta change the preferences manually once a year. Don’t know if you can change but it’s such a minor inconvenience I never bothered to check.