The free option with libraries was somewhat limited because libraries rarely had more than a small number of copies of a given book or newspaper.

Saying people with higher income bought for convenience is understating the situation. If you needed timely access to books or newspapers that were in high demand you often had to buy out of necessity.

I'm not sure how a similar thing would work the internet. How do you limit the number of people that can use the free version? If you don't the people who could pay largely won't.

The only idea I've heard that might work is to just make it free for everyone, but put a tax on something that correlates somewhat with use, and divvy up that tax to the sites based on their traffic.

That then raises questions like what to tax and how to divvy it up since simply dividing it proportionally to traffic probably would not work well (Richard Stallman has suggesed such a system, with the split based on the cube root of popularity).