While I'm sure plenty of sites do, a huge venue like NYT has so many readers there is bound to be HN users among them that will post anything remotely relevant. Most times I see something on a major site that I think would be worthwhile for HN, it has been posted long before I saw it.

I know there are "services" that can post your site on various social media, but I don't know how prevalent they are or if it can be easily detected.

(Opinions are my own) They don't. A site like HN isn't in the top 10,000 referrers and no one has ever heard of it; traffic from this exact post is barely a blip. There's no business value.

Teams responsible for backlinks would post on established social media in a very transparent way (from official accounts, etc.)

Besides everyone here has JS disabled, four ad blockers installed and uses a text-based browser. If I personally made the call they would never have employees (or paid contractors) post here of all places.

It's never stuck out to me as being overrepresented in posts here, but I've never thought about it either. Full disclosure: it's one of the main news sources I follow.