Why not go back to a client hosted MUA (thunderbird etc) like we used to do before gmail.

For my part, because I need continuous access to all my mail/contacts/calendar from several mobile devices (phone+tablet) plus several computers.

Before gmail and good mobile devices I used mutt on a shell account to get access constant access but that was only mail + contacts.

I haven't seen a mobile device that doesn't do IMAP/CalDav and/or Exchange. Why do you need to go to a website on a mobile device to read your email?

No, but most webmail providers dont offer calDAV + cardDav or exchange on any inexpensive account.

The problem is not the device but the providers.

CalDav, CardDav and IMAP should have these all sync together, no?

Yes but no eu company offers it as a inexpensive package.

If you absolutely cannot find what you're looking for then nothing stops you from just firing up your own email server and putting whatever UI you like on it.

That's what I did a while ago and I haven't looked back, using Gmail/Yahoo/whatever is a miserable experience in comparison.