The USD is probably the most well-behaved fiat currency of the last century, so that’s basically the least interesting example, but:
USD is inflationary. Ceteris paribus, a rational actor would generally prefer to hold something deflationary. The (epsilon-) no-arbitrage argument suggests that the frequently raised “but then no one will ever invest in things” objection doesn’t actually make any sense.
USD is highly controlled (but not by you). This is a technical and legal issue, not a fundamental problem with fiat currencies in general, but it’s impossible to ignore. Bitcoin sidesteps it entirely. What actually drove me to try Bitcoin is when a paperwork error by the state comptroller led to them withdrawing several thousand dollars from my bank account with no warning. I eventually got it back when they realized they made a mistake, but Wells Fargo still charged me “legal fees” and the realization that I really didn’t have control over my money was uncomfortable. With bitcoin I can use my money however I want, whenever I want, and no one else can. No paperwork, no ACH/fed clearing, no delays, no surprise fees, no one stealing money out of my account with impunity. That’s only the tip of the iceberg for how much the legacy financial system sucks, and most of it can’t be fixed with a software upgrade.
Fiat could theoretically work as well as bitcoin in terms of convenience and reliability, but it never does in practice and there’s no way to avoid governments instituting worse policies later on.
I also don’t think most Bitcoin supporters want the government to get rid of USD. It’s about adding choice, not forcing everyone to do something different.