I'm probably missing most of your point, but wouldn't the fact that we have inverse problems being applied in real-world situations somewhat contradict your qualms? In those cases too, we have to deal with noisy real-world information.

I'll admit I'm not very familiar with that type of work - I'm in the forward solve business - but if assumptions are made on the sensor noise distribution, couldn't those be inferred by more generic models? I realize I'm talking about adding a loop on top of an inverse problem loop, which is two steps away (just stuffing a forward solve in a loop is already not very common due to cost and engineering difficulty).

Or better yet, one could probably "primal-adjoint" this and just solve at once for physical parameters and noise model, too. They're but two differentiable things in the way of a loss function.