The trademark issue is pretty cut and dried: a trademark holder must show due diligence in protecting its trademarks or else it loses control of the trademark permanently; as a result, a trademark holder has every incentive to send nasty messages to anybody they might have reason to believe might be seen by future courts as infringing upon their trademark, lest they be suspected of not protecting it well enough. (Basically, the law incentivizes trademark holders to be assholes & go overboard, which is why Disney sues day care centers for having Disney-owned characters on murals.)
As for linking: you may not remember, but the court case regarding the legal status of links was a *big* deal, and old media companies are likely to still be pretty sore about it. It was only twenty years ago that the question of whether or not deep linking was a form of infringement; similar mechanisms (like the use of frames to wrap ads around other people’s sites) were ruled against. It doesn’t really matter if the NYT is planning to make a bot: they didn’t want to give up on suing Google & anybody else who does deep links, and now that they have to, they have every intention to prevent that verdict from being generalized to slightly different circumstances.
Furthermore, you can make the argument that use of the API might, in certain cases, lower the number of ad views they get or screw up their progressive paywall system. While I’m not convinced that either system is making them bank in the way that they might hope (and both are easily & regularly bypassed), they have a financial incentive to label and shun anything that might conceivably be used to circumvent these mechanisms — particularly if the product is not commercial.