If you self host as an ALTERNATIVE to the 3rd party you have all of the same problems - more because you know about them, and the 3rd party can make all these claims you can't verify until they fall over with a "load balancer misconfig" story you also can't verify.
If you self-host redundantly to a 3rd party you have no special benefit (it does the same thing) AND the additional cost of a redundant infrastructure.
Why not just have redundant 3rd parties (so-called "multi-cloud") if you can't or won't trust your 3rd party.