Epistemic status: speculating mostly from first principles with no expertise.
I don't think the NYT specifically will be captured. I believe the NYT is currently censored in China. Further, I think that most likely the NYT is quite far from being uncensored; substantial changes in editorial policy would be required. For instance, the NYT is willing to publish opinion peaces from democracy advocates in Hong Kong. The NYT also probably doesn't have an existing customer base in China (partially due to this censorship). This is unlike Hollywood. For instance, Marvel movies are quite successful in China, and Marvel (Disney more generally) targets its movies for Chinese audiences and to appease the Chinese censors already. Further, the NYT generally seems to follow a center left or maybe somewhat progressive political stance and China is generally quite unpopular among this cohort at the moment.
While I suppose it would be possible for the corporate managment to greatly alter the editorial policy to gain access to the Chinese market, I think this would cause a huge backlash in the US unless the typical center left and/or progressive stance on China changed substantially. This backlash would include the journalists working at the NYT and I think the attitudes of the journalists greatly shape the overall stance and policies of the NYT.
So, for the NYT to be captured I think that a considerable shift in the standard US center left and/or progressive political views on China would have to occur. This would also have to be accompanied with radical changes to the editorial policy (no publishing of pro-democracy opinion pieces from Hong Kong for instance). This isn't entirely unprecedented. For instance, a NYT opinon editor resigned after an editorial by Tom Cotton was heavy critiqued (at least in left wing circles). This is despite the fact that the NYT will publish editorials by the Taliban! See also "I can tolerate anything except the outgroup" by Scott Alexander.
It's hard for me to see how such a large political shift would occur.
As far as the capture of other large US media institiutions, I think a good indicator would be if they are currently fully censored. If so, probably the institution is unlikely to be captured anytime soon - there isn't a clear feedback cycle which would lead to capture.
EDIT: also, I commit to responding to comments on this answer for the next 2 weeks.