Prior to the rise of advertising, almost all sites were 'independent'. They'll be around for a long time after the end of Internet advertising, because they're run for love, not money.
And none of those sites carried breaking news or the AP wire, at least not legally. Or had sports scores (ditto). Or showed streaming video other than self-produced content in 240 x 160 "QuickTime postage stamp theater" format. Or paid anyone to write content for them. Or provided social media capabilities (vital to the ubiquity of the Internet, whether you personally like/use them or not). Or did much of fucking anything other than be personal projects or part-time blogs that ran until the proprietor got a job/spouse/kid and realized it was an unsustainable time (and bandwidth cost) investment. All that would be left is e-commerce sites; personal sites where the creator can handle technical duties and pay the cost of hosting (remember, no ad-supported WordPress!); 100% sponsored sites (which would thereby lose all credibility of independent thought); big corporations that could afford making "loss leader" websites or sustain the costs of being subscription-only (also as bad); a tiny number of donation-only sites like Wikipedia with enough notoriety to sustain themselves; and some government pages funded by your tax dollars.
I loved the era when you had to install WinSock or MacTCP to use your college's Internet connection. Browse the Wayback Machine from 1996 and you'll get warm fuzzy feelings, but remember that this was when the Internet was a nerd phenomenon like Usenet, not a global force for easy information dissemination and democratization of media. To return to it would be the death knell of the Internet in all functional ways.
Advertising may be annoying. But it is what fueled the growth of the Internet into what it is today, and I personally don't see celebrating the death of sites like Ars Technica, Longform, Foxtrot Alpha, Jalopnik, The Onion, Kotaku, TheForce.Net, Grantland, Slate, ESPN.com/SI.com, or pretty much any other site on the web that I currently enjoy for free. Insert whatever other site here that you enjoy reading and you do not currently directly pay for.Your mileage may vary, but I don't think "the rest of the world will celebrate" as you seem to believe.