You don't have to setup an entire company, you just have to find other peers who want to coinjoin with you.
Which is practically not an option, and even if it is, you gain much less privacy in comparison with JoinMarket, wherein there are thousands of BTC in liquidity.
And still, you can't expect from a regular user who wants to have their coins coinjoined to find peers. Like, what's more useful, to simply be a taker or a maker in JoinMarket and have it mixed in a couple of hours max, or reach o_e_l_e_o, n0nce, witcher_sense, 20kevin20 etc., and have them do a coinjoin with me?
The point of owning the private coin is so you can verify the coordinator is not performing a Sybil attack on your non private coin, there's no trust at all.
Are you daft or something? So, SBF can setup a coordinator and do coinjoin with one of their peers, and they'll be considered private? Are these totally acceptable to the default coordinator now? A few pages before you'd have them denied. Won't the third party (who's by the way a chain analysis company) trivially notice it's their coins?