People - usually suit-wearers - have been making this spurious claim for decades, but it doesn't hold water.

The largest of the finest print reminding you that it's 'sold as is' (or more encompassing variants that might continue '... with no warranty for fitness of purpose') means that liability remains in the lap of the purchaser / consumer / operator.

(This has been a source of immense frustration over my career - where such people have assured me that they have 'recourse' (it's always vaguely described) by spending money on proprietary products & services, rather than opting for functionally equivalent or superior free options.)

I think your third paragraph is implying a distinction (or conflating the difference?) between LLMaaS's and self-hosting publicly available models.

If it's just where it's hosted that provides the legal insulation then things like OpenRouter would give you that. (But again, I suggest that it would not.)