Is it just me, or would a giant lookup table fails much weaker tests that you can throw against it. (for instance: just keep asking it to do sums until it runs out)

Well presumably the lookup table can have steps you go through (produce this symbol, then go to row 3568), with state as well, so it’s more like a Turing machine than a single-shot table lookup.

That's a motte-and-baily really. If one starts out with a LUT and then retreats to a turing machine when challenged, that is. If our friend in the Chinese room is in fact permitted to operate some sort of Turing machine, I'd make a very different set of inferences!