Earlier quoted context omitted.
> point the MX record to whatever hosting provider you want You can even have a hybrid solution where incoming mail goes directly to your self-hosted server and (some) outgoing mail is relayed through a third party.
What would be the advantage?
1. Control. A third party can change anything about the service any time they want, and if you don't like the change they made you're screwed.
2. Expectation of privacy. Because I am not contracting with a third party, the government cannot argue that I have waived my right to privacy. (As a practical matter of course this matters not at all. If the government -- or anyone with the right technical skill and access -- wants to read your email they will. But if push ever comes to shove in a court of law it could matter.)
3. Spam filtering. I think the whole industry is doing it wrong. The Right Way to filter spam is to use your outgoing mail as ground truth for what is not spam. I have a custom spam filter that I wrote based on this idea and it works like a charm. No Bayesian analysis needed. I don't even look at content at all. Just the headers are enough to achieve >99% accuracy.