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A thing already known to computer scientists, but still useful to remember: as per Kleene's normal form theorem, a universal Turing machine is a primitive recursive function.
Meaning that if an angel gives you the encoding of a program you only need recursion, and not unbounded search, to run it.
The claim as stated is false. The standard notion of a UTM takes a representation of a program, and interprets it. That's not primitive recursive, because the interpreter has an unbounded loop in it. The thing that is is primitive recursive is a function that takes a program and a number of steps to run it for (this corresponds to the U and T in the normal form theorem), but that's not quite the thing that's usually meant by a universal machine.
I think the fact that you just need one loop is interesting, but it doesn't go as far as you claim; if an angel gives you a program, you still don't know how many steps to run it for, so you still need that one unbounded loop.