Earlier quoted context omitted.
They've pretty much all tried having comment sections, and removed them. Turns out, it's kind of hard and the incentives discourage it.
This is not what I mean. What I mean is that they (CNN, FOX, MSNBC, NYT, WSJ et caterva) could, e.g, set up a Mastodon server and give accounts to their staff and at least encourage cross-posting. They could give free accounts to their subscribers. There is so much that they could to send a strong signal that they are against Twitter/Facebook as the single place of public discourse. But of course they don't do any of…
Attempting to clone Twitter, even if only for their staff and subscribers and/or in a federated form, opens them up to the exact same issues that motivated them to get rid of the comment sections. There would even be the additional problem that they'd need to advertise the thing. The software being available off the shelf doesn't matter, because the software is not the only hard part.
Of course NYT are more interested in buying Wordle than in trying to "combat" the big tech oligopoly. They also didn't try to compete with, say, Ma Bell or Clear Channel, but they have been a big player in the word game space for longer than the personal computer has existed.