Emergency funds for some of them are not to be taken to invest, but rather emergency funds for their living needs when they are pressed by a disaster that occurs, be it a flood, earthquake or the death of a relative. So they can use the emergency funds for that, instead of selling assets/bitcoins they own. In fact, what you explained is not wrong, but the term might be better used as cushion funds or reserve assets that they can take so they don't miss the stage of their purchase.
Emergency fund is not recommended for investing in bitcoin but no bad to use it because put emergency fund in bitcoin have potential in the future price keep increasing than hold in pocket only, not really difficult for withdrawing emergency fund when saving in Bitcoin due put or hold in exchange wallet and when emergency happening to us we can sell and withdraw to bank account without take 10 minutes. No one can't predicting when happening with something urgent for us and my opinion better put emergency fund in bitcoin investment and seems the same when have parent hold or save their money allocated for children education. Hold in the bank can make it double although we start investing when our children under two or three years and take at seven or ten years later. Its very important allocated our fund for children cost but effective when saving or put in Bitcoin.