There’s no way to do a FHE style system with branching unless all branches are indistinguishable from one another. All branches must execute in the same amount

There’s no way to do a FHE style system with branching unless all branches are indistinguishable from one another. All branches must execute in the same amount of time, otherwise statistical analysis will leak information. Forget Turing completeness for a second, that’s not actually what you want for a FHE style system. You want computational indistinguishability which can’t be achieved without side-channel resistance.