Why does the proof of undecidability of $A_{TM}$ require the universal TM to take input $\langle M,\langle M\rangle\rangle$?
Why not give it the input of $\langle D,w\rangle$?


What's $w$? The machine $D$ is given some input, which it interprets as the coding of a Turing machine $M$. It's not given any $w$: only $\langle M\rangle$.

Imagine instead that you're given a function $f$ which is defined as $f(x)=x^x$. I ask you to calculate $f(10)$ and your question is a bit like asking, "Why does it have to be $10^{10}$?  Why can't it be $10^y$?"