That is not how it works. In difficult times, people will be more interested in assets that can act as store of value. So they will go after bullion, United States Treasury bonds.etc. Cryptocurrencies are being regarded as a high-risk asset, and therefore the interest in them will be rather muted during hard times. But as the market matures, this situation may change.
That's certainly true, mate. Cryptocurrencies have a higher risk than traditional Fiat because of their wild fluctuations in price across the market. At least we have a solution for this, bringing people the convenience of Fiat with the security and reliability of Blockchain technology. The solution is called a "Stablecoin" which retains the stability of Fiat while relying on Blockchain technology for security, speed, and reliability. People who want to mitigate the risks of crypto's volatility, could simply convert their crypto holdings into a stablecoin of their choice. The process is instant, easy, and secure. This is much better than directly cashing out to Fiat, subject to the control of central banks and governments worldwide. With a decentralized stablecoin like DAI, you can enjoy the benefits of Fiat without ever worrying about this.
Of course, crypto will never replace Fiat since it's decentralized by design. Fiat will always be patronized by mainstream businesses and companies, while actively supported by the government. The fact that Fiat is backed by the full faith and credit of the government, gives it some legitimacy over crypto itself. That's how people think nowadays, so they'll continue to use Fiat over crypto regardless of its disadvantages (like inflation, government control, etc). Anyone has the choice to support one type of currency or asset from another. Safe-haven assets like Bitcoin or Gold will always exist because they provide an "exit route" to the traditional banking system of today. People who want to secure their money in times of turmoil, will quickly resort to these assets. But it's evident that a store of value can never replace Fiat because it's a completely different thing. It's like comparing apples vs oranges. Let's hope crypto becomes more popular, so that it could become as good as Fiat in the future. Just my thoughts