This may sound reasonable but isn't at all how newspapers are run. You can absolutely buy an ad in the New York Times criticizing the New York Times. Within reason of course, as you said the are private entities allowed to take on any customers they want, but in general newspapers hold journalistic integrity as an ideal and will allow most things as long as they aren't defamatory, unethical or downright illegal.

The ad sellers and the journalists are normally separate and will not interfer much with one another's work. It also helps that they never say no to money. I don't know about the New York Times specifically but similar things have happened many times in other newspapers, and there is such a thing as a paid editorial. Those are usually clearly marked as such, but it's basically the same thing.

(However, there may be other reasons why you might want to go with a competitor instead, and not pay the newspaper you hate $100k.)