This comes up every time there's a NYT article. However, note that the management and the journalists are different people and the fact that th website of the NYT itself is reported as having trackers is a proof of this.
That's clear and not being disputed. When people mention workers calling out management, it's sort of in a praising way. They're commending management for at least allowing workers to call them out.
Many institutions would not allow that to happen, so it's good to point out the Times when they do.
They do it in the open. They say they do it. They tell you they know it's bad, this thing they do. And they do it anyway.
It doesn't make them better than others to pay lip service and not change. If anything, they lead by example and it makes them worse for having done so.
Shop lifting is a crime, but hey, everyone does it, so watch me pocket this candy bar. And some beer. And a TV. And maybe I'll just grab some money from the cash register. And yeah, I think I'll steal the assistant manager's car to get away. See? This is just how the world works!
Upper management can interfere but usually this is done via the editor and usually only when they really don't want a story to run (e.g. they think it'll bring too much heat or it accuses one of their friends of paedophilia).