For web pages, you could imagine a sort of universal "like" button which pins the page locally and bookmarks it. I think it's less about incentives and more about convenience -- people are happy to help out if it's not too much trouble for them.
I thought the last 20 years on the Internet have shown us that "humanity is noble, and good" isn't true, and we should assume "evil and/or lazy".
So yes, they will "unpin any site as soon as it becomes a resource drain" unless they're feeling particularly charitable - but that's exactly how it works with torrents, too! Generally you set up rules like "seed until a maximum up/down ratio is reached for that file, then don't bother", "restrict uploads to a certain maximum speed and/or certain maximum amount of data per day", that kind of thing. I figure you'd basically have an automatically-managed cache of limited size that you'd use to hold the stuff you were liking-to-pin, and stuff that was judged to be no longer worth seeding would get kicked off, as would old/unpopular content if the cache had filled up and you were shuffling in new content.
So unpinning anything that becomes a significant resource drain is also - at least in the relatively-short-to-medium-term - fully compatible with other people being willing to "like" your site to pin it, if not perpetually, than for an extended period of time.
so you're disincentivized from using the service
It's a huge oversimplification to describe people in just those terms. People are complicated and so are their motivations, and it's hard to tell in advance which new things will be successful or not.
For example, I once worked for a guy who hadn't heard of open source and, when I described the concept to him, couldn't wrap his brain around why so many people would take the time to write and maintain a bunch of software and then just give it away. This is not the work product of "evil and/or lazy" people.
We know these two things happen, consistently. We have failed to account for them in much of the foundations of the Internet - it's the assumption of benevolent cooperation. Which was great for ARPAnet, but simply doesn't work in the wild. We have 20+ years of proof.
We cannot continue to put our heads in the sand and ignore that, because for better or worse, the Internet has become a major force shaping society.
The idea that naivete translates into mental health is certainly fascinating, but I believe the colloquial translation for that is "ignorance is bliss".
If most people use smartphones to consume content on metered internet connections with limited battery life, where do they pin it such that it's available to others?
(around 65% of Americans between 18 and 50 have PC/Mac laptops- see https://www.statista.com/statistics/228589/notebook-or-lapto...)
Here's a graph of average hours of computer usage per US household in 2009:
https://www.statista.com/statistics/187622/average-hours-per...
Let's assume these figures are useful as a first approximation of the number of hours of laptop usage per laptop owner in 2018. I'm not saying it's a perfect match, just that they're likely to be better than figures you or I would pull out of the air.
Taking the centre of each band, assuming "more than 10 hours" means "10 to 16 hours", and excluding the "no computers" band, gives a mean of 4.3 hours per day. So each laptop is active roughly 18% of the time.
That's more than I expected, but it still means you're going to need a lot of people to pin the content before you have a 99% chance of at least one copy being online at any given time.
Edit: As far as I can tell, with 18% uptime you need 15 copies for 99% reliability (assuming no correlation between the online times of the various copies, which is optimistic - in reality there will be strong daily and weekly cycles).