... For how long should people understand the importance to have bitcoins for emergencies such as this one?
Okay let's assume the lady has bitcoins on her non-custodial wallet but she still needed to sell it for fiat because the hospital doesn't accept BTC. What if she tried to trade it on a centralized exchange and the platform suddenly froze her account for some reason? That's still going to be a problem. How about P2P platforms? A lot of them requires bank accounts for trades so the same issue.
It's not really bitcoin's problem but the point here is that there is always a risk as long as people continue to use centralized platforms.
Even when we use a DEX, are your Bitcoin sales not going to be paid into your Bank account? As far as I know, for now, the DEX I have used also pays directly to your Bank account and we are back to the same problem except you know someone that is willing to pay you by hand directly.
I would guess that you could send your money to coinbase or binance or whatever pays in your fiat and then you could cash hat out, in this case we are still talking about having bitcoin as your emergency money but I would highly suggest you do not do that.
My understanding is that the money you put into bitcoin should be money that you do not plan on spending at all, it should be there for a decade if it could, even longer if it could stay that way. If you have money that you plan on spending in case of emergency, then life is filled with those type of emergency stuff and that doesn't change the fact that you would end up with something terrible in the long run, like price dropping and then you cashing out because you have to.