You pay enough for monthly internet service as it is.
None of
that money makes it to any of the people who offer websites.
That would be as dumb as NBC or ABC charging per show that you watch.
Not dumb. "Scary" perhaps.
As you allude to, the issues
are similar: Both websites and broadcast television are paid-for by the viewers' willingness to watch advertising commercials
and to be influenced by those commercials with regard to their purchasing decisions. In both cases, the traditional means of delivering commercials can now effectively be avoided (for television, DVRs can skip ads; for websites, ad blockers work well), and even when folks don't skip over commercials (with a DVR or ad blocker), it is becoming clearer that more and more people are not allowing themselves to be influenced by advertisements.
That all makes the value of your viewership worth less. If something is worth less, then you're going to pay less for it, right? Well same with advertisers. As commercial avoidance increases, and as commercial effectiveness decreases, advertisers will need to find new ways of activating purchases. That means that they'll either take their advertising money away from broadcasters and websites, or they'll force broadcasters and websites to make advertising more invasive and less avoidable.
In the case of a website that doesn't want to make advertising more invasive and less avoidable, the only alternative to remain revenue neutral is to replace lost advertising dollars with subscription fees. Will people pay? I doubt it. I think too many people are going to flock to the places that remain "free", even though those places choose the other alternative: more invasive and less avoidable advertising.
So the end result is probably going to degradation of the online experience -- but at least websites will still mostly be "free".