Bitcoin can do without fiat and likewise fiat can also do without Bitcoin since they both represent money in different form. Money is anything that is been accepted by people to use in exchange of goods and services. When there was no fiat currency peoples were still trading using other means of money. Bitcoin just came into existence mean while before this we were using fiat.
The difference between both currencies is that one is more valuable than the other because Bitcoin was created due to to weakness of fiat to safe its users from financial freedom. In one word both currency are independent of the other.
I will not agree with you on the part that "bitcoin can do without Fiat and Fiat can also do without bitcoin " how? And how many things have you used bitcoin to buy in your locality? Bitcoin is only an alternative currency to the Fiat currency, and that means bitcoin can't stand alone because you can not use it to buy things in your local market. But Fiat can be used to buy things both online and offline but bitcoin is only online. Therefore, it is only Fiat currency can be used to buy things in my locality but bitcoin can do things without Fiat currency. Bitcoin cannot stand alone but Fiat stand alone.
This is the best example of bitcoin and Fiat in the market exchange 👇
I think that for bitcoin itself, fiat is unimportant, as noted above. But for us, unfortunately, fiat is still needed since bitcoin is not widely accepted everywhere, and we cannot use it fully when trading various goods. In my country, there is a strict law that prohibits the use of bitcoin. Therefore, I am forced to exchange it for fiat in cases where I need to purchase something.