Grep works just fine, despite implementing non-deterministic finite state machines. Monte Carlo simulations are behind nuclear weapons (where they were invented), weather forecasts, financial trading, etc. Las Vegas algorithms like randomised quicksort also diminish no one's responsibility.
In principle, you can run training and inference on neural networks completely deterministically. But I don't think that makes any difference to responsibility. (To make them deterministic, you obviously have to use pseudo-random number generators with fixed seeds, but less obviously you also have to make sure that when you merge the results of parallel runs, the results 'merge' deterministically. Deterministic parallelism is an extremely interesting field of study! Or, since we are only talking about principles, not what's practical, you could just run everything in series.)
The problem with LLMs is that they are complicated and their actions are hard for humans to predict or reason through. Complexity is the bane of responsibility: if you have a complicated enough system (and a complicated enough management structure involved in producing that system), that's where responsibility goes to die, unless you specifically work to establish it.
In this case, employing LLMs is no worse than employing humans. If upper management gives bad instructions and incentives for lower level employees, we tend to pin the responsibility on upper management.