> I obviously don't think anything is wrong with using alternative currencies, but I'm not sure that separation of currency and state can happen the same way separation of church and state has. Even if the government doesn't control the issuance of currency they will undoubtedly have an outsized influence on whatever currency they choose to support.

I honestly don’t see the big difference: the people, by voting on representatives, choose both which religions and means of payment the government accepts (assuming a democracy, of course).