I just looked up the share structure; didn't realise the publicly traded shares only appoints 1/3 of the board. Still their second best option is start buying up competitors and going ahead with purging NYT from their training set. That might well end up a worse option for NYT, as they won't stop LLMs from gradually intruding on their space and the moment OpenAI or other LLM providers own major publishers so they don't need to depend on scraping, they lose any leverage they currently have.
They won't need to. Most don't have enough money to survive a prolonged round of lawsuits, and the potential damages are limited. The only real leverage is taking their models out of circulation and cutting their training set and that leverage only exist for the large publishers.
I'm not convinced it's a given it will. If it becomes necessary to license, owning the large publishers will be leverage and allow locking competitors out unless you have a portfolio to cross license.
OpenAI alone has a market cap that'd allow it to buy about as large a proportion of publishers of newspapers and books as they'd be allowed before competition watchdogs will start refusing consent.
Put another way:
If I was a VC with deep pockets investing in AI at this point, I'd hedge by starting to buy strategic stakes in media companies.