This opens up a big discussion. If it's separate from fiat, then why do we measure bitcoins in dollars, euros, yen and not the opposite? We could do both, but we would never measure a dollar in sats. How is it supposed to work without fiat, since it's historically shown that the government has to control the money of the society in order for it to operate properly?

To answer the OP's title, I personally think that it can't work without fiat existence, because I don't believe in anarchism nor do I think that governments would regulate a censorship resistant type of money.