So why do ads follow you even after you bought the product? Well the ads are still less than 1% effective (products sold / by ads shown), so for the average company (not Amazon) it is just not worth the engineering effort for a < 1% savings on your remarketing budget. Secondly, this is more speculation, but for many products the chance of buying a product given you have already bought one (e.x. a spare or a gift) is still higher than chance for a random person. As such the ads are still effective.

So what is the deal with the NYTimes? They a huge popular publisher (especially amount rich people) already command high ad prices through traditional adverstising. The value of a remarking ad (to the advertiser) is largely independent of the site that it is shown on. So The New York Times would naturally have fewer of these ads because the adverstiser gets more ads/$ on other sites. A diffrent site like a local paper might have a diffrent story. However remarketing ads have never been a large percentage of ads shown so from a revenue stand point publishers care a lot less about them.