Yes, they can and that's how it's set up. Each community makes their own rules and can choose who participates.
It's not Reddit. It's the sub that made the decision and I'm not sure how it would be possible for Reddit the company to deal with sub level rule complaints and appeals.
Leaving aside everything else wrong with that, that would be trivial to abuse, especially with the help of sockpuppetry but easily enough even without that.
It doesn't matter what you post, just the association with that sub is apparently enough.
If you're at the point where you have been vetted and allowed to post on r/Conservative, you've gone way past mere "association." This isn't like some board game forum where you can just create an account and start posting. r/Conservative (probably with good reason) has a long and very active vetting process before you're allowed to post there, and only posts that conform to their ideology stay up. So getting banned for participating is a little more than just "guilt by association."
If this guy had disclosed his conflict of interest, he would just have been an obsessed crank and even as a moderator, that's his right. But when he didn't, I'd say it was large-scale manipulation, and it's clearly in Reddit's interest to not allow this sort of thing (especially now that they're selling all their data to AI companies).
I'm not sure, as in this case it seems to rise to Defamation + Trade Libel/Commercial Disparagement. So it may go beyond being simply unethical.
If one is squatting on a valuable forum name, then the moderators should be themselves subject to a standard enforced by Reddit.