Mandrill or Mailgun are used for sending out transactional emails [?], not your domains' regular email flow.
What you're looking for is something like MailChannels which you can configure as a smarthost, and route all outgoing email through their mailservers.
Last time I check they're basic plan was $50/mo for up to 500.000 emails/mo.
You could also use SpamExperts Outgoing filtering (pricing depends on the number of domains that are going to be sending out mail). You can find the cPanel integration instructions for the outgoing smarthost here, plus there's a cPanel addon aswell.
I know these are actually antispam gateways, but that's essentially the kind of service you need to route all your domains' outgoing mail through.
I've been using Mandrill for over 8 months and like it a lot. While it does send out the transactional emails from my website via API, I also send all of my desktop client email and even my phone via Mandrill SMTP.
Why can't we continue using Mandrill to send for all our domains' mail via my server and not just our transactional mail from my server?
Mandrill is a completely neutral platform for emails of all kinds. Although it's primarily designed to support transactional emails, you can send any legal, non-spam message through Mandrill, including bulk or general day-to-day emails.