My biggest problem with using any other app than Gmail or InBox is GMail's category inbox - Primary, Social, Promotions, Updates, and Forums. I have InBox set to alert me only with Primary and the rest - which Gmail does a fantastic (actually, perfect) job with sorting out for me - are all not worth being reminded of every time I receive one.
Again, Outlook comes closest with its Focused vs. Other mailboxes. Again, it learns what is critical/important over time. What I love about InBox (and Outlook), in addition, is the ability to "snooze" a message to come back later / tomorrow / this weekend / next week / whenever I want. I' couldn't manage with any other email app without these features.