Put all likes (and dislikes!) on a public append-only ledger, and allow people to buy and sell them freely and publicly. Advertising and social proof rolled into one - after all, money is the best social proof we've been able to invent.
Users who care about organic original types of things are then free to filter out bought likes and look at just the ad-free graph.
To draw on your analogy to explain the problem here, you may notice that despite the USA having lobbying, it also still has corruption.
A distributed consensus network would be an ideal medium for this hypothetic model, as there would be no single intermediary able to fiddle with the interaction counts, and everybody would be free to crawl the resulting social graph using whatever algorithm their user agent provides.
Explain to me what non-paid interaction you are thinking of that I cannot do to someone's post if they hand me a $10 note, but can do if they didn't.
Subjectively, I would consider a single interaction that merits a $10 cash transaction effectively "organic": proof-of-human-work, as it is. Anyone thinking otherwise would be free to add parties caught in off-the-books transactions to their personal blocklist - as is the case with ad blocking today. Entities compiling blocklists could then be rewarded accordingly, and in turn withdrawing trust for a blocklist considered "corrupt" would be a one-click operation.
The long answer: the commonly leveraged argument that it's impossible to conclusively verify that the contents of a distributed ledger correspond to some sort of "base reality" is simply moving the goalposts. It is already impossible to independently verify that a private database on a cloud somewhere is consistent with itself, and thus that it corresponds to any reality at all - you have to blindly trust that the organizations involved would prevent conflicts of interest. And as long as they can get away with it, they're going to tip the scales in their own favor - which is the reason public ledgers face such vehement pushback from entrenched authorities.
To continue this line of thinking, society operates just fine on a shared consensus about what "reality" is, even though in the grander scheme of things it's impossible to conclusively prove anything at all - except maybe some abstract mathematical statements. All that blockchain technology is trying to do is leverage those to introduce increased transparency and granularity into the mechanisms of establishing societal consensus - which already operate on a scale where classical methods of public oversight are insufficient.