German law stipulates that ToS requiring a call (or letter) to cancel when you subscribed online are void (§ 309 Nr. 13 BGB limits what restrictions can be placed on the mode of communications). Cancelling via their contact form, email, or even SMS is totally fine as far as the law is concerned. I don't see how the NYT could refuse your cancellation via their contact form. Maybe nobody's bothered to sue them? (IANAL)

Deepl translation of the law: Even to the extent that a deviation from the statutory provisions is permissible, the following shall be invalid in general terms and conditions: [...] 13. (Form of notices and declarations) A provision by which notices or statements to be given to the user or to a third party are bound a) to a form more stringent than the written form in a contract for which notarization is required by law; or b) to a form more stringent than textual form in contracts other than those referred to in point (a); or c) to special access requirements