Iirc TST doesn’t deny anything about, or really ever mention CoS. CoS doesn’t like TST because of the same reasons you listed (basically they don’t like being conflated or associated with TST, cus their super cereal).
I personally love the absurdity of TST, and heavily believe in their mission/philosophies. Lucian Graves has some issues, but overall they track overwhelmingly on the side of good. Am also a lifelong gothy metal kid so the aesthetic also pleases me.