I used to be in the "no webmail" camp until I started using Fastmail. I now find myself keeping a Safari pinned tab to their webmail and using it as my primary email interface.

Also speaking of mobile apps, Fastmail actually supports push notifications for email in Apple Mail. I'm not aware of any other third party email provider that's ever done that (I believe Yahoo! has support for that but Apple did that themselves when the iPhone first came out).