The diagonal lemma and the existence of quines already show that you don't need specific support for self-reference in your language, because any sufficiently powerful language can formulate self-referential statements.

In formal language terms, it would be more accurate to say that any sufficiently powerful (ie: recursively enumerable, Turing recognizable, etc) language must contain some means of producing direct self-references. The existence of the \mu node in the syntax tree isn't necessarily intuitive, but its existence is a solid fact of formal-language theory. Without it, you can only express pushdown automata, not Turing machines.

But self-referencing data structures within a single Turing machine tape are not formally equivalent to self-referencing Turing machines, nor to being able to learn how to detect and locate a self-reference in a universe being modelled as a computation.

In fact, UDT uses a quined description of itself, like in your proposal.

I did see someone proposing a UDT attack on naturalized induction on this page.