Is there a more intuitive proof of the halting problem's undecidability than diagonalization?
It may be simply that it's mistaken to think that someone would reason their way to this argument without making a similar argument at some point prior, in a "simpler" context.

Remember that Turing knew Cantor's diagonalisation proof of the uncountability of the reals. Moreover his work is part of a history of mathematics which includes Russell's paradox (which uses a diagonalisation argument) and Gödel's first incompleteness theorem (which uses a diagonalisation argument). In fact, Gödel's result is deeply related to the proof of undecidability of the Halting Problem (and hence the negative answer to Hilbert's Entscheidungsproblem).

So my contention is that your question is in a sense badly founded and that you can't reach the Halting Problem without going past the rest (or something remarkably similar) first. While we show these things to students without going through the history, if you were a working mathematician it seems unlikely that you go from nothing to Turing Machines without anything in between - the whole point of them was to formalise computation, a problem many people had been working on for decades at that point. 

Cantor didn't even use diagonalisation in his first proof of the uncountability of the reals, if we take publication dates as an approximation of when he thought of the idea (not always a reliable thing), it took him about 17 years from already knowing that the reals were uncountable, to working out the diagonalisation argument.

In reference to the "self-application" in the proof that you mention, this is also an integral part of Russell's paradox (which entirely depends upon self-reference), and Gödel's first incompleteness theorem is like the high-powered version of Russell's paradox. The proof of the undecidability of the Halting Problem is so heavily informed by Gödel's work that it's hard to imagine getting there without it, hence the idea of "self-application" is already part of the background knowledge you need to get to the Halting Problem. Similarly, Gödel's work is a reworking of Russell's paradox, so you don't get there without the other (note that Russell was not the first to observe a paradox like this, so prototypes of the diagonalisation argument has been around in formal logic since about 600BCE). Both Turing and Gödel's work (the bits we're talking about here that is) can be viewed as increasingly powerful demonstrations of the problems with self-reference, and how it is embedding in mathematics. So once again, it's very difficult to suggest that these ideas at the level Turing was dealing with them came a priori, they were the culmination of millennia's work in parts of philosophy, mathematics and logic. 

This self-reference is also part of Cantor's argument, it just isn't presented in such an unnatural language as Turing's more fundamentally logical work. Cantor's diagonalisation can be rephrased as a selection of elements from the power set of a set (essentially part of Cantor's Theorem). If we consider the set of (positive) reals as subsets of the naturals (note we don't really need the digits to be ordered for this to work, it just makes a simpler presentation) and claim there is a surjection from the naturals to the reals, then we can produce an element of the power set (i.e. a real) that is not in the image of the surjection (and hence derive a contradiction) by take this element to be the set of naturals who are not in their own image under the surjection. Once we phrase it this way, it's much easier to see that Russell's paradox is really the naïve set theory version of the same idea.