Both of the subscription sites you mention have ads in both their paper and electronic editions. They receive a significant proportion of their revenues from advertisers. They don't have high net profit margins.
What makes you think they don't rely on advertising revenue to stay afloat?
I was curious about the details, so I looked up The Economist's 2018 annual report. They get 17% of revenue from advertisements, while 60% of revenue comes from subscriptions. Their operating profit is 13% of revenue.
Right, but of course without ads they would save some cost as they wouldn't need people to sell ads, they could get rid of a bunch of ads-related overhead costs etc.
If the operating margin on the ads part of the business is less than ~25%, then they could break even without ads. Of course, breaking even is worse than making 35MM profit each year.
What makes you think they don't rely on advertising revenue to stay afloat?