This remedy would not affect the free Internet.
Here is why: Google is not going to suddenly start charging consumers for search because it now has to pay a lump sum transfer to newspapers once a year. That lump sum transfer is not going to enter their pricing calculus. It would completely destroy their business model if they said that, instead of making money off of advertising, which is the heart of their business and what they've been doing for a decade or more, they're going to start charging for search. They're not going to do that.
It's the same calculus for Facebook. If Facebook has to write a lump sum cheque once a year that doesn't vary with output and doesn't affect their marginal costs.... As a micro-economist, I can tell you that it's not going to change their pricing optimization and it's certainly not going to cause Facebook to deviate from its model of free social media in return for monetizing the eyeballs in advertising.
I hope this answers your question at a few different aspects. The part that is sticking with me is whether it is going to cause the Internet to somehow no longer be free. It is not going to affect the pricing decisions of the platforms at all.
Search will forever be free. Social media will forever be free and a lump sum transfer at the end of the year will not affect that.