Sort of moot anyway. If statements can approximate any function, most programming languages are effectively turing complete. What's important about specific architectures like transformers is they allow for comparatively efficient determination of the set of weights that will approximate some narrower class of functions. It's finding the weights that's important, not the theoretical representation power.

"Consider this: can an mlp approximate a sine wave?"

Well, yes - we have neutral speech and music synthesis and compression algorithms which do this exceedingly well...