So your second intuition about proof of identity is better, but the minute you invent a foolproof identity system and it gets widly used (by fiat or even voluntarily), you give a tremendous amount of power to the issuer/manager of it, as they basically hold the power of also removing/altering your public identity at their whims. You also make it harder and harder to remain anonymous and preserve your privacy, two things that might seem useless in a functional democracy, but unfortunately democracies are not perfect and they can turn into totalitarian systems where the wet dream of dictators is exactly this: having total control over every individual, especially the ones who would dare signing petitions against them.
This is by far not something humans have solved systemically, we have the technical solutions to build such systems, and governments tend to push to implement them, but as someone who cares about privacy/censorship resistance I do my best to build alternative, opt-in, systems to go around them.