Increasing the number of 3rd parties authorized to access your email accounts increase the potential attack vectors for malicious people to utilize. If you only use GMail's interface, then you need to trust Google to protect your account. If you grant Acompli access to your account, now you've got to trust two companies with that responsibility. It's basic cost/benefit in terms of what you get from providing a 3rd party with that access - and in a day when email is still one of the more critical achilles heels in resetting account access, it's advisable to keep that access as limited as possible.
Have you tried Inbox? Inbox is nice but I am not sure why there is no easy way to delete an email in the app. Mailbox has implemented that well - half swipe -> archive, full swipe -> trash I guess?