Does table salt dissolve in a liquid metal the way it dissolves in water?

The answer is yes, but only slightly, and the reason your searches kept flipping the question is itself the explanation.

Salts and metals are both good solvents for their own kind of thing, and they are poor solvents for each other. Molten salt is ionic, so it dissolves ionic compounds readily and dissolves metals to a small and interesting extent. Liquid metal is metallic, so it dissolves other metals readily — that is what an alloy is — and dissolves salts only slightly.

So the two systems are not symmetric in interest. Metal-in-salt is technologically important, so it is measured and written about at length. Salt-in-metal is a small number that mostly matters as a contaminant, so it is measured in narrow contexts and reported in places that do not surface in a general search.

The short version of what would actually happen: mostly it sits there, with a small amount going into solution and a large amount forming a separate layer, usually floating because salts are typically less dense than the metals you named.

There is a mechanism worth naming because it makes the small solubility less mysterious.

When a salt does dissolve in a liquid metal, it often is not sitting there intact as ions the way it does in water. The metal can reduce the salt's cation — a reactive metal will simply take the chlorine and swap places, producing a different salt and releasing the other metal into solution.

So "does it dissolve" and "does it react" blur together in these systems in a way they do not in water. That is another reason the literature is thin and scattered: the answer depends on which specific salt and which specific metal, and half the interesting cases are chemistry rather than solubility.

The flux example is the one that would have answered this immediately if I had thought of it. I was searching for a solubility figure when the everyday industrial evidence is that the two phases separate cleanly enough to be a purification method.

From the practical end, this behaviour is used deliberately and you can see it in any foundry.

Molten metal is covered with a salt flux precisely because the two do not mix. The salt layer floats, keeps air off the metal, and collects oxides and rubbish out of the melt, and then you skim it off as a separate solid layer.

That process would not work at all if salts were freely soluble in liquid metals. The whole technique depends on the answer to your question being "barely".