I can see it making sense if you're already using the mail servers of the company that made the app (gmail for example) since they already have unrestricted access to your login info and mailbox anyway.

I guess otherwise I'd rather it be my phone regularly polling my mail server over my connection than a third party regularly polling it using theirs. I can see it being a popular option for folks on expensive/very limited data plans though. It doesn't take a ton of bandwidth to poll a mail server, but checking every 5-10 minutes it could add up.