Indeed. I wasn't envisaging that someone would use an ISP's mail server and then run their own mail server AS WELL.
I suppose one might do that to ensure one had a backup server in case the ISP suffered an outage (with the whole mail database, not just a secondary MX) ... but of one is trusting an ISP's mail server at all one should probably trust the ISP to provide redundancy and backups.
[Or, rather, if one's ISP doesn't provide redundancy and backups then one probably shouldn't trust their mailserver.]One might run other sorts of server, though, and rely on the ISP for mail.