I don't think "wordle" was a popular name for the game before it got popular on the internet, so a trademark makes sense in this case.
Of course Wordle is just Lingo but online. And Lingo is just televised Mastermind with letters instead of colours. Mastermind is an extension of Bulls and Cows and if you go back further I'm sure you can find another game like it.
There are some very big broadcasting companies that have put a lot of money into the "guess the word" genre and the New York Times probably know that they can't take down every game with similar rules.
The best they can do is find infringement of their trademark and pursue those who infringe upon it, or risk losing the trademark all together. They could also license the trademark, of course, but a blanket cease and desist letter is cheaper than negotiating a contract.