Though, like what you have said it takes time if in case that government around the world will consider bitcoin as alternative system, so far with those digital types of currencies crypto/bitcoin is not one of the top priorities.
only time can tell if that will happened, let us enjoy that they are co-existing to benefits more people.
Cashless payments are already running now, because they are considered very effective, and there is no need to carry cash anywhere. For Bitcoin
to be accepted as a payment alternative in all countries in the world it takes a long time, because for now in some countries, Bitcoin is still considered
a threat, because it cannot be controlled. But with time many countries will realize the importance of accepting Bitcoin. The conclusion is that
it is impossible for Bitcoin to exist without fiat, because the government will not allow this to happen. The most reasonable thing is that Bitcoin
and fiat co-existing, if this happens, many people will feel the ease of conducting financial transactions.
Bitcoin's market cap would have to be so huge that it could act as a unit of account, seriously, probably at least 10 or 20x from where we are. If you value stuff in Bitcoin, you are also running into the threat of prices getting manipulated by whales. Bitcoin won't exist alone, there will be other currencies that it somehow relates to, hence there is always room for manipulation.