These small providers I had the experience with, they worked fine without a mobile verification.
- e.email: seems like a big company, has community, storage and free email
- wail.ch: Swiss based, email and storage for free
- safeswiss: Swiss based, a free secure email provider
- Runbox: though this is a paid mail, they really don't require mobile for verification
I have used all of them for long days (more than half of a year), never asked me to input mobile, and email just works. so I suggested.