Digital-only subscription revenue in 2017: $340 million
Digital advertising revenue in 2017: $238 million
Obviously some digital advertising is shown to print subscribers logging in too... but if we ignore that for simplicity, without ads your NYT digital subscription would be 70% more expensive.
At $195/yr. for digital-only, it's already my most expensive subscription by a significant amount. If it were 70% more, or $331.50/yr, I imagine subscriptions would drop by a lot -- that's a lot of money for news.
So... that's presumably why there are still ads. Gotta pay the reporters somehow.
On the other hand, the New Yorker bugging you to subscribe is just lazy programming, and I hate it too.
[1] https://www.nytimes.com/2018/02/08/business/new-york-times-c...