I would be happy if you publish and sell a book (booklet?) containing selected articles that appeared on the site (without comments)

Some articles are long and thought provoking and deserve a good slow reading while cuddling under a blanket.

I have an account in amazon which allows me to pay in one click.

Recently I noticed that two of my favorite sites had a gif from amazon which said something like "Shalom Chen! Click here to pay to 2$ to the owners of this site"

After enquiring abit into this mechanism and checking that the 2$ will really reach those sites, I now click weekly.

Yep, But the credit system works because I prefer to pay 2$ weekly rather than 100$ yearly.

And they are getting the entire sum. I asked the two site owners.
But they are paying Amazon for this service.

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i think there might be some who would see your definition of "your role" as strange, to say the least.
free market economy is based on all doing what they can to preserve their own interests, within the limits of law.
this means that your so called "long view", with everyone doing their part is more of the social engineering socialist/communist concept, rather than conform to free market forces. this would be commendable, if we were talking about some social policy, with benefits you can not quantify monitarily. however, the relationships between you and the newspapers is such that if you imagine you can "play along" and tolerate something to reach a balance, the first player (in this case another newspaper) who fucks you over and takes advantage of that does the right thing, gaining competitive advantage over the other newspapers (at least for the time being), and potentially drags the other newspapers into ad wars, with you having nothing to do about it (remember, you were trying to be nice). by taking care of your own interests and using every legal (nothing immoral about it) means in your disposal to make the surfing experience more enjoyable, you would be, broadly speaking, transmitting an actual signal to the market about your likes and dislikes, causing the different newspapers to actually reduce the amount of ads, as to eliminate the need for you to block them.
the newspapers role is to make money - by selling ad space, by selling content (to Walla! for example), and by subsciption fees.
the advertisers role is to make money by making you buy stuff through advertising, placing their advertising in effective cost/benefit locus.
your role is not to make money, and is not to pay money. it's to read content. and the newspapers and advertisers will adjust to that, since it's a client driven market. there are very low/non existent switching costs for you.