There's been a debate for a while now about this. Clay Shirky, Walter Isaacson, Jakob Nielsen, Norman Hardy, Kevin Kelley and others have expressed arguments for and against(mostly for) micropayments. But there's yet to be a success story, most early implementations failed(BitPass, Ezcash, Peppercoin) but that was before cryptocurrencies. Now there's a new crop(Steemit, 21.co?, etc) that are trying again. Interesting to see how this will all pan out.

I want to pay a fixed amount per month, and distribute that money over the websites I visited, weighted by the time I spent there.

A micropayment system tied in to browsing history might make this possible. Otherwise it would require essentially a syndicated subscription system, which is server side, but would need enough content producers on board to make feasible, and they probably wouldn't be able to agree.

Sounds like https://flattr.com/