Why don't we see more Kings Indian Defenses in top level chess? One of the most interesting openings, in my opinion, is the Kings Indian Defense (KID). 2-4 decades ago it used to be very popular at the top level with Bobby Fischer and Garry Kasparov using it regularly. Even a few years ago it used to be quite popular with Radjabov and Nakamura having it in their repertoires. 

Why has it started to go out of fashion at the elite level, and especially with the Neural Network engines on the rise? Leela has been able to use it quite effectively. Nakamura said in an interview that the Kings Indian is lost for black. What are the drawbacks of the openings that prevent it from being seen at the top level?

The Interview is from 2015 Sinquefield cup after Hikaru obliterated Wesley So with the Kings Indian
It is simply not as solid as other openings, mostly because it cedes too much space, and especially with the advent of computers: White has simply found more ways to press black. 

In a 2017 interview, Magnus Carlsen did not think too highly of it. 
"If he had wanted a worse position, he would have played the King’s Indian," Carlsen said. 

Nevertheless, it still gets played at the highest level from time to time, probably mostly as a surprise weapon. Here is a recent search of Mega 2020 (I previously searched incorrectly, and have edited this). This is from 2015-2020, sorted by rating first, and then date. You can see that despite the quote above, even Carlsen has played it.