You have $100k. Put all of it in the stock market. The stock market drops by 40%, your investment is worth $60k and you withdraw $20k. The market goes back to 100% and now you have $67k in your portfolio.

If the market dropped by 80% your portfolio would be gone entirely. Meanwhile if you had enough emergency funds you would have kept everything.

The worse the emergency, the higher the ROI of the emergency fund. Since there is no upper bound for how bad an emergency can be the theoretical ROI of an emergency fund is also unbounded.

In this comment the definition of an emergency is a stock market drop combined with unexpected unemployment.

To me, an emergency fund is a hedge against various different risks like these (I am guilty of not being prepared for all of them):

1. A pandemic. For this I was ready even before we thought it's indeed possible - I have funds in several different banks (debit cards) that allow me to not have to visit a bank physically for several years (even if some of the cards expire in the meantime, or one bank suddenly goes bust)

2. AI glitch, identity theft, a bank holiday (this happened to me actually some years ago) that blocks you from accessing banks (and investment brokers if you have nowhere to withdraw your funds) for several weeks/months - for that you hold paper cash, physical gold and cryptocurrencies stashed at different physical locations.

3. Natural disasters - again some combination of diversified physical and virtual funds

4. Legal trouble - yes, even if you are not guilty you could get your funds blocked.

I presume you are from the US, hence the purple goggles ;) (I hope I'm not offending you)