Not to completely plug my own work here, but I also wrote about this for a slightly more mathematical audience (and uhh, a much shorter post): "There are exponentially many vectors with small inner product"

https://lmao.bearblog.dev/exponential-vectors/

For those who are interested in the more "math-y" side of things.

For what it's worth, I don't fully understand the connection between the JL lemma and this "exponentially many vectors" statement, other than the fact that their proof relies on similar concentration behavior.