The integration is extremely easy to setup [2], and solves the deliver-ability issue. Most of these services also take care of the DKIM and DMARC setup, so it is much easier to get things right. And they all offer debugging tools if something isn't sending or returning correctly.
On the other hand, they are not in privacy abusing position that Gmail is, and I also don't have to worry that they will close my account with no recourse, as Gmail once did to me.
The biggest downside is no email forwarding. There is no way to do email forwarding using a service, since they only send mail from approved senders. (There are Bad Idea solutions, such as rewriting headers with SRS. Don't.)
And of course, I need to pay for sending. Mailgun used to have a generous free tier, and Amazon SES is decent at $1/10K mails, but things add up. I chose Postmark for many reasons, but the pricing of different companies is worth looking into if you are sending enough mail. (Sendgrid and Mailchimp Mandril, for example, have a unlimited per deliveree charge, that could conceivably be cheaper for some use cases.)
OT, the only reason that I support PoP3 is for Gmail: The only way to "check an external account" to write and send mails in Gmail is PoP3.
[1]: postmarkapp.com [2]: https://postmarkapp.com/support/article/832-can-i-configure-...