Would NYT publish scathing remarks about itself, or publish leaked internal data on itself? If not, I don't think such a thing should be expected of twitter either.

The NYTimes has investigated itself from time to time and publishes remarks that reflect poorly on themselves. Last example I can remember is the Caliphate podcast scandal

https://www.nytimes.com/2020/12/18/world/middleeast/caliphat...

Yea doesn't the NYTimes have a section pretty much devoted to retractions? I'll see them in their morning news letter and I think it's pretty common at most news organizations with integrity to understand that sometimes they'll get things wrong like we all do, and just be candid about it and post retractions where necessary.

Do you mean Twitter in it’s entirety is like a respectable newspaper that should edit all their content for correctness and propriety?

The BBC certainly broadcasts criticism of the BBC from time to time. Here’s one of the more trivial examples (easier to bring to mind because it’s amusing): https://youtu.be/MBWfCV4PjyU

Crude, and hilarious to watch those two keep straight faces. Have I Got New For You has very regular criticism of the BBC too, slightly higher brow but entertaining.