> instead of users submitting a paywalled version of the article they could instead reference one that seems to not have this issue
People often do this when the site is known to have an impenetrable paywall, e.g., WSJ.
But the NYT's paywall is quite easy to bypass. It usually only appears if you've accessed more than the monthly limit's worth of articles, and after that you can just use a clean browser instance.
Bear in mind that the NYT is generally regarded as among the premier news sources in the world. (This is not to say that it's to my personal taste. It often isn't, but that's a separate issue.) But its articles, particularly those with an opiniative or analytical flavour, will often be preferred due to this brand perception that the NYT has built.
Where there is a better article on a topic, people can and do point to it here and encourage the moderators to update the URL.
But the HN community would never want a lower-quality article on a topic to be given preference due to it having no paywall. If that article is the best one on the topic, as long as the paywall can be bypassed, that is the article HN wants listed.