It would make fiat worthless because nobody would want to have any, kind of like how no one under 40 reads newspapers any more now that we have the Internet.
That's not entirely true. Fiat has an unfair advantage which is legal tender laws. If you run into someone's kid and get sued the court will order damages paid in fiat. If you have none you need some. Same with tax liabilities.
This is why the U.S. Constitution mandates the States make nothing but gold or silver coins legal tender. If paper can become legal tender, as it is now, then it can have market value even if it wouldn't normally. This gives those in control over the fiat, banks and politicians, exceptional power.
This is why historically governments that desire to be corrupt prefer its citizens not use gold/silver. Bitcoin has the potential to rein in and regulate government as gold can, although fiat can still have some value as long as it remains legal tender.