Nobody thinks a perfect market, or perfect anything is possible. It's just a theoretical concept. None of the imperfections you mentioned can be solved by a state or any other form of coercion.
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America is not a "free market" capitalist country. I would say it's of the lack of a free market that enabled the wealthiest to capture that wealth that would otherwise have been more evenly distributed. There is an enormous amount of regulation in the US (and other countries) designed to make it nearly impossible to compete with the established actors in the market.
Facebook polls are neither secret nor verifiable. Facebook knows how everyone voted, and voters cannot verify that Facebook is reporting the correct result. Those problems can theoretically be solved with cryptography (assuming the voter has full control of the device used to cast their vote), but not in a way the average voter can understand and verify.
Another problem with online voting is that malware can infect a voter's device to spy on how they vote and/or change their vote. I don't think there is a solution to that problem.
Online voting is a terrible idea. The only way to have secure elections is to publicly count every ballot by hand. If you think that's too expensive, you think electoral democracy is too expensive.
It wouldn't be an arms race, it would just be treating everyone equally, albeit equally poorly. Unbalanced sexism looks more like an arms race as the those who are in the victim group will try to reverse the roles, and if they succeed, the new victim group will try the same. By far the most common justification for sexism seem to be perceived unfairness and sexism by others.
I don't find any gender's "locker-room talk" funny and would rather not hear it at all, but if one gender's version of it is on Netflix, the other's should also be.
I think casual sexism is harmful, but I think unidirectional casual sexism is likely much more harmful.
How do you define "feminized nonsense" and "gender politics"?
Are women (in science or other areas) underrepresented as subjects of Wikipedia articles relative to their notability (as defined by Wikipedia)?