There is no need for government to back anything. Currency has historically been and can be again privately issued. For instance, Tether. If they back USDT with BTC instead of fiat trash, with a certain level of auditing and proof of reserves, won't you trust it?
Would I trust it? Maybe for a small amount I didn’t care about losing.
But what is the point? Why would anyone want to receive Zimbabwe dollars over USD? That is what USD is compared to BTC - it’s Zimbabwe dollars.
having something to trade that reliably loses value is not required for economic activity - if it were, why isn’t Argentina1000x more productive than the United States?