But modern computers let you send them instructions entirely in a Turing-complete language, with no requirement that you first know how to unroll them into a static circuit (constant-bounding every loop). That seems like a bigger difference than "oh sometimes you get out-of-memory errors".
As far as I can tell it is possible to homomorphically compute a UTM. Since it's possible for a homomorphic circuit to decrypt and reencrypt its own ciphertexts as part of the bootstrapping principle it should also be possible for a circuit to fully decrypt certain values (outputs). Basically, if V = E_K(V') is the homomorphic ciphertext of V' under the primary key, K, and O is an output key, the primary circuit can compute V* = E_O(D_K(V)). An output circuit D_O(X) decrypts values encrypted with the output key, e.g. V' = D_O(V*).
A UTM is an unbounded tape T of encrypted bits, an encrypted state S, a circuit U(S, B) = (S', B', V') representing the state machine of the TM with V' as an output to control the next step, and the circuit for D_O(X). After each evaluation of U the state S is replaced by S', the bit at the current position on the tape is replaced with B', and if V = D_O(V') is 0, move the position on the tape left, if it's 1 move right, and if it's 2 then halt.
A bounded-length Turing Machine can be represented as a finite state machine (with the computing power of a standard physical computer) without leaking any information about changes to the tape by (T', S', V') = FSM(T, S) where the entire tape T is replaced by the output of the circuit at each step and D_O(V') is 1 for continue or 0 for halt.
