Okay but that example is just one conditional branching. Turing-completeness requires that you be able to e.g. keep looping back until some condition is met.

If you want to build arbitrary functions like you've described, it reduces to the question of whether you can "unroll" the function, given its inputs[1], into a (stateless) circuit. But that would still IIUIC just get you primitive recursive functions (i.e. those where you can know the upper bound on the number of loop iterations for any loop before it starts), not the general recursive functions you need for Turing-completeness.

Anyone know how you pull off general recursive functions from just addition and multiplication?

[1] and in FHE you don't know the inputs either, which is another kink.