Yes bitcoin is independent because it is decentralized but in the price of bitcoin are also dependent on fiat currency .
Bitcoin is dependent on Insane amounts of Electricity to run it's network,
Electricity that has to be Purchased with the Local FIAT
So your statement is correct:
"Bitcoin is dependent on fiat currency for it's existence"
No Fiat, No Electricity, No Bitcoin.
Bitcoin doesnt consume any energy because its online and totally passive , it only uses electricity if you have a mining hardware and you are mining a bitcoin .
Also , we can still have an electricity even without fiats because we can use natural energy such as solar , water and wind .
In conclusion , it does not really matter at all if bitcoin is independent or not because what matter the most is the usability and purpose on why it was built .