Yeah, don't put everything in one basket. As bad as fiat money appears to be right now, the benefit of having a savings account in a bank is usually that your deposit is guaranteed by your government.
Generally, the case for Western countries, but I'm not sure this is the case for most of the countries in the East. Also, your savings is generally only covered by certain banks, and if you have multiple bank accounts belonging to the same group, you can potentially get caught out. So, that's worth noting if you're relying on that guarantee that the government does. It's these incentives that keep people storing money in the bank, since being your own bank definitely comes with its risk, and a certain level of knowledge to keep it safe. Not everyone, is comfortable with that.
It's the same even in emerging countries, so I know for sure South Asian and Southeast Asian, all banks that get the licence from the central bank benefit from a uniform deposit guarantee scheme. It's been tested in one or two countries already, and we've famously bailed out banks multiple times, post second world war anyway. When Japan came in the 1940s most countries here weren't independent so the scene was full or private banks and money disappeared overnight -- that was actually what led to deposit guarantees by independent governments.
Most people won't reach their deposit limits in their lifetime anyway...
But you're right, I've always thought most people aren't prepared for the responsibility that comes with owning Bitcoin -- and not knowledgeable enough to realise it.