Running in-house email servers does provide more control and wider customization as well as tracking and compliance for small to midsize businesses (SMB). But they require specialized on-site IT staff as well as the need to manually manage and support both hardware and software. A third-party hosted email service matches many of the advantages of in-house email without the expensive initial investment. The ability to manage the number of users, access the latest security protocols, and enjoy ease of connectivity and deployment of hosted solutions makes it a viable and competitive option.
That's a big difference, with "inbox zero" requiring an email client with great archiving that works over multiple device types. Meanwhile, at the other end of the spectrum, the personal information managers need something more like Microsoft Outlook, with excellent search capabilities as well as a good storage contract on the hosting side because these types of inboxes are often tens of gigabytes (GB) per user.
One of the most important compatibility factors to consider with email is the mobility question. How often do your employees need to access email via mobile devices? That's an important issue because most email hosting providers deliver some kind of web client usable as a default inbox. Almost all of these can be accessed via a mobile device, so if your employees don't need to access their emails on the road that much, then such mediocre clients are probably fine.