It would be extremely hard to issue cash as non government assuming the government doesn't want cash to exist, it would basically require a centralized entity that could be trivially shut down by the government.
My point is that there is nothing stopping bitcoin from being used in a paper cash like way, any more than any other value carrying token.
Bitcoin without government support? Possible, but inconvenient and dangerous. Bitcoin with government support? More nice things.
Currently we are in the "with government support" world, more or less. I hope this continues.
Even then, almost all PoW are tuned in a way to pump the value and enrich the early people, not fairly distribute tokens or make the crypto functional as a currency.