Again, if you aren’t using port 25, you don’t need an MTA. MTAs are for sending mail from one mail server to another. That’s not what you are doing. You don’t need to be running your own mail server. You are sending mail to an MSA (Message Submission Agent). Mailgun is the MTA that will handle transferring your message to your destination MTA.

If you were submitting mail to your destination server over port 25, you would need an MTA.

But, as you stated, you are submitting mail to an MSA using port 465 or 587. You need an MUA.

This terminology is important, because it determines what role the software you’re using plays and how you need to configure it.

There are plenty of tools that act as both MTAs and MUAs, because their roles are very similar. If you want to use the right tool the right way, I am telling you, with 100% certainty, you should be looking for an “SMTP client”, “email client”, or “mail user agent” (they mean the same thing). You do not want a relay server, and you do not want an MTA.