Can a Turing machine compute the outcome of any machine that is less powerful than a Turing machine?
Pick any model of computation, say a class of Arbitrary Automata $\mathcal{A}$, so that $F_{\mathcal{A}} \subseteq \mathrm{RE}$, i.e. this model is sub-Turing-complete.

Since TMs are an admissible numbering and $\mathcal{A}$ is a numbering of some set of (semi-)computable functions, for every $A \in \mathcal{A}$ there is a Turing machine $f(A)$ with $F_A = F_{f(A)}$ -- and we know that this compiler $f$ is computable.

By the existence of a universal Turing machine $U$, we get that $U(\langle f(A) \rangle, x) = A(x)$ for all $x$. Applying the s-m-n theorem gives us a TM $T$ with $T(\langle A \rangle, x) = U(\langle f(A) \rangle, x) = A(x)$.