Managing mailing lists or list servers is not something I’ve done before. I believe some of the key features people use include customizing every single link for tracking purposes, built-in un-sub features, double opt-in support, and probably other stuff I’m not aware of.
There are open-source tools created for this purpose, at least some of which are listed at Awesome-Selfhosted.
However, I’m not aware that Mailgun can replace list servers, such as GNU Mailman (or Majordomo).
If you do not need your list mails to be sent out in high volume super fast (I’ve tested https new sessions on $5 server at 60-100/second, not sure how that translates to mail), I would choose Vultr, and even go to a higher plan depending on the cost difference of using a provider such as Mailgun. However, I am a person who goes through conf options carefully and would just jump in to learn mailing list and list server packages.
Note that I would also contact Vultr support to see if they have any issues with a server that sends out mailing lists. I suspect that as long as the lists are clean with double opt in and unsub links at the bottom, they won’t have any issues with it. I’ve read through their TOU and most if it I found very reasonable given they are in the VPS business.