The thing is that these legal threats are usually empty, as far as the chances go of actually winning. There might be more cause for the concern in the UK or other countries, but in the US it's very hard for a public figure to win a libel suit, even if the reporting is sloppy, or even if the reporting contains substantial false information, as long as it wasn't done on purpose. That was established in
New York Times Co. v. Sullivan.
If the reporter is unlikely to have to hand money over via a judgement or settlement, that leaves the cost of hiring lawyers to defend the case. For an individual, that cost can still be ruinous, which is why so-called SLAPP suits really are a big problem. But an organization like the New York Times gets sued all the time. They have in-house counsel and paying them is part of the cost of doing business.
In theory, the cost of defending more libel suits could still be problematic in aggregate. But the proof is in the pudding. The New York Times is constantly publishing negative articles about all sorts of people. Just looking at the front page and limiting to investigative reporting (not coverage of specific news events, and not editorials) of targets that could theoretically sue (not e.g. Putin): today there's an article on the owner of 4chan; yesterday there was one on Trump; the day before, one on a large hospital chain (which "spent years cutting jobs, leaving it flat-footed when the pandemic hit"); the day before that, an article naming several extremists attending a Young Republican Club event. And of course there are many other articles that people could take offense to; Sarah Palin recently lost a libel case against the NYT for a 2017 editorial.