Considering bitcoin investment as emergency money doesn't seem a bright idea because this will mean that once you need the money you should sell it a market value and sometimes it could be below your entry point.
The exit strategy is a long time movement because you must wait for the right price to convert your money into fiat so you won't be able to have access to your funds right away.
Emergency money should be in fiat.
It is true that I agree with you that emergency funds should be in fiat money. Why? Because if for example there is our family who are sick and must be immediately taken to the hospital so if we save the emergency fund in the form of bitcoin we will bother selling bitcoin first and then can get the fiat money. So it's best to have two savings one for emergency funds in fiat money and investment savings in bitcoin.