Search engine companies pay for the software and infrastructure necessary to crawl the world wide web in order to index all the accessible content (subject to directives which may be included in web pages to deprecate indexation and crawling), and to assess (and rank) the accessible content for (primarily) relevance. Then they provide access to the indexed, ranked content via search engines. The frequency with which results are chosen (popularity) in turn affects rank.
That service is what makes most of the web accessible to us. Without the search engine companies, we would know only the domain addresses (very, very few pure IP addresses any more) of web sites passed along to us, discovered by us, or linked by sites we already know.
That service is also what makes most of the web usable to many - perhaps most - content providers. Except for the obviously well-known site owners, without search engines it would be an extraordinarily hard lift for anyone else to get any visibility.
Search engine companies and many other web site owners generate ad revenue by serving ads (including ads in their own pages) and collecting information to verify the number of impressions served and the number of click-throughs. (Note that an impression is counted regardless whether a user actually reads anything on the served page.) Those are the basis for billing the ad owners. If the search engine provides a useful service so that many people use it, its revenues improve. If a web site provides useful content so that many people use it, its revenues improve.
So "big media" sites are in the same situation as every other web site owner attempting to generate revenue from ads: produce content people want to read. Note that this is not the same mission as providing content they want people to read.
Without search engine companies, "big media" web sites, along with everyone else without other sources of revenue to pay for their web presence, would most likely collapse. Search engine companies are already providing a free and valuable service to every web site owner.
The first thing to sort out in this debate, then, is whether the service provided by search engine companies to media companies (steering web traffic) is worth more or less than whatever it is media companies think they are owed for content. I can guess that without search engines serving up multiple results for the issue of the day, people would be much less likely to go to more than one or two web pages; thus the total numbers of ad impressions would fall off; thus that part of ad revenues would fall off.