It was a bit difficult to grasp at first, but it clicked after realizing it's all primes up to 37, not just 37. Kind of a neat fact, and I enjoyed reading this. Thanks for posting!
37 is the median factor, it doesn’t mean that 37 is the actual second prime of 50% of numbers. The OP probably missed “median”, I know I did
"We see that 37 is the prime where roughly half of all numbers have something less than or equal to 37 as their first prime! So we’ve proven that 37 is the median second prime!"
Can you elaborate on "it" in this sentence? What is "all primes up to 37", when you phrase things in an easier to understand way?
I can't tell if you're suggesting the word median is being used in a misleading way? But it's basically the definition of "median 37" that half your numbers are "up to 37".
Now I understand that with enough composite numbers, the chance that their second prime factor is 37 OR BELOW converges to 50%, which is pretty neat.