How does someone get such a job? It’s not like the nyt puts out an online app, screens thousands, then picks a random individual. It’s always going to be through personal networks.
If you want to get your crossword published by the NYTimes, all you have to do is write a good one and submit it.
I've met a guy who has a couple of crosswords published by them. He didn't have personal connections with anyone, he's not anyone special, he's a regular guy with a regular job. What did he have to do? He liked crosswords and tried writing them, then submitted them. They got accepted and published, that's it. There's no shadowy cabal of crossword writers secretly running the world.