Is there a more intuitive proof of the halting problem's undecidability than diagonalization?

Question

I understand the proof of the undecidability of the halting problem (given for example in Papadimitriou's textbook), based on diagonalization.

While the proof is convincing (I understand each step of it), it is not intuitive to me in the sense that I don't see how someone would derive it, starting from the problem alone.

In the book, the proof goes like this: "suppose $M_H$ solves the halting problem on an input $M;x$, that is, decides whether Turing machine $M$ halts for input $x$. Construct a Turing machine $D$ that takes a Turing machine $M$ as input, runs $M_H(M;M)$ and reverses the output." It then goes on to show that $D(D)$ cannot produce a satisfactory output.

It is the seemingly arbitrary construction of $D$, particularly the idea of feeding $M$ to itself, and then $D$ to itself, that I would like to have an intuition for. What led people to define those constructs and steps in the first place?

Does anyone have an explanation on how someone would reason their way into the diagonalization argument (or some other proof), if they did not know that type of argument to start with?

Addendum given the first round of answers:

So the first answers point out that proving the undecidability of the halting problem was something based on Cantor and Russell's previous work and development of the diagonalization problem, and that starting "from scratch" would simply mean having to rediscover that argument.

Fair enough. However, even if we accept the diagonalization argument as a well-understood given, I still find there is an "intuition gap" from it to the halting problem. Cantor's proof of the real numbers uncountability I actually find fairly intuitive; Russell's paradox even more so.

What I still don't see is what would motivate someone to define $D(M)$ based on $M$'s "self-application" $M;M$, and then again apply $D$ to itself. That seems to be less related to diagonalization (in the sense that Cantor's argument did not have something like it), although it obviously works well with diagonalization once you define them.

P.S.

@babou summarized what was troubling me better than myself: "The problem with many versions of the proof is that the constructions seem to be pulled from a magic hat."