It's a work in progress, Bitcoin can't be a Fiat alternative without fighting some war with centralised entities first and this can take years to accomplish and we shouldn't worry ourselves over it, my belief is bitcoin will make it happen anyways.
Well, not everyone would think like that since I believe both fiat and Bitcoin should exist at the same time. They can't eliminate fiat and switch to Bitcoin since only fiat is being controlled by the government therefore, they can have it while owning a Bitcoin will remain in your own wallet and you are free to use it as you may. So it's just up to the people if they want fiat or Bitcoin, as long as there is a merchant who could accept Bitcoin or fiat if someone wants to withdraw.
So choosing between fiat and Bitcoin is still on a big debate.
In actual it depends on time and on the people and on the technology of future but in this time it seems that usage of bitcoin is more convenient for people as nowadays many of the purchases are done via online services and for fiat people feel it difficult to go through complicated processes of online payment processors. If the usage of bitcoin is allowed by the authorities without any restrictions then people will like to choose bitcoin to use for their daily transactions.