Not exactly. While it is possible to become a Guardian 'supporter', and contribute financially in exchange for ad-free browsing after you identify yourself by logging in, you are not free from tracking by third-party ad companies as you browse their site.
The parent comment you are responding to said "I think the business model would generally incentivize much more user-friendly operational decisions than an ad-sponsored approach does." and while this gives you a superficially cleaner browsing experience, ad companies continue to track you as you browse the Guardian's website, harvesting data which can be used to manipulate you elsewhere on the web.
The Guardian could choose to disable all third-party trackers for logged-in paying supporters, but they do not. For privacy conscious readers, the only way to achieve what you want is to install an ad-blocker and a browser extension such as Privacy Badger. Contributing to the paper provides little or no privacy benefit, though it's always a good idea to support quality journalism, so if you do it, do it for that reason alone.
Be sure to include your subscriber/membership number in that email, which you can find in your profile https://profile.theguardian.com/
Perhaps it's a return to tried and tested model rather than a switch to a new one?
I believe Facebook is overvalued, but comparison to the value of a dog park is not exactly fair either.
What I was trying to convey was that a social network doesn't necessarily need to move the mountains of money that Facebook does. Diaspora* is an example of a free open source social network platform. Of course it still cost people a lot of time and money to build and host it, but nowhere near Facebook levels.
I also think the business model of running specifically a social network as a profitable business is likely to lead to "perverse incentives". Making a profit isn't necessarily opposed to helping people connect (e.g. OKCupid's business model of being free to all but selling advantages to power users), but it takes both the willingness to align your business model with your users' interests and the fortitude to stick to that approach even though other approaches could be more profitable.
One is a great place do be if it wasn't for all piles of steaming crap left around by irresponsible users.
The other is somewhere to exercise your dog
You could run a social media platform instead as a service for users funded by donations, in the vein of Wikipedia. This way you don't need to convince most people to pay. You only need a small passionate few who keep the service going for everyone else. I think the business model would generally incentivize much more user-friendly operational decisions than an ad-sponsored approach does. And since this is a digital service, the operational cost can be scaled proportionally to the donations received.