The last two are straightforward. The proof relies on a result called the Golod-Shafarevich theorem that gives a criterion for a group to be infinite. Golod and Shafarevich proved this a long time ago (1964). Moreover, if you look at how Golod and Shafarevich used this criterion, it's the same way it's used in the proof: They apply it to some Galois groups that appear in number theory, prove these are infinite in certain cases, and deduce that there exists an infinite tower of number fields with some surprising properties.

Much more recently (2021), Hajir, Maire, and Ramakrishna figured out how to apply the Golod-Shafarevich theorem to a slightly different Galois group to produce an infinite tower of number fields with some even more surprising properties. This is used in the new proof. It requires very slightly modifying the construction of Hajir, Maire, and Ramakrishna to produce the fields needed in this proof, but the explanation of how to do this takes only a paragraph in the human-written summary. (The explanation is more laborious in the original AI writeup).

The relation to Ellenberg-Venkatesh is more indirect. This is where "in retrospect" comes in because this work was not cited in the original AI proof. This has to do with the next step of the proof, after you construct the number field, you need to find many elements of this field with the same norm to produce many vectors of the same length. To do this, the proof uses a pigeonhole argument which uses small split primes of the field (constructed via Hajir, Maire, and Ramakrishna's argument) to construct many ideals. By the pigeonhole principle, you can guarantee two ideals lie in the same class. When two ideals lie in the same class, you get an element of the field. You can rig things so these elements all have the same norm. Ellenberg and Venkatesh had an argument which also used the pigeonhole prnciple to guarantee two ideals lie in the same class to produce elements of the field. They were working on a different problem so their argument was slightly different, but similar.