Both are possible, it is just that since we have no way to control the events that happen around us, being too aggressive in our investment strategy can be very detrimental, if we find ourselves facing a difficult set of circumstances.
So the safest path is to build your emergency fund and only then begin to invest in bitcoin, however such an approach has an opportunity cost and you may miss buying bitcoin at an attractive price, so I can understand why you may prefer to take a hybrid approach.
I prefer to buy Bitcoin at an attractive price, namely when the price has fallen or is still in a fairly low area historically for several months. However, the investment approach as you say is also very good because everyone is required to have an emergency fund even though the fund is not used often every day. But I actually prefer to buy Bitcoin first, even in a smaller amount, rather than preparing a larger emergency fund because I like holding Bitcoin even if the amount is not that much. Because preparing an emergency fund also takes time and it could make us miss the best price on Bitcoin so that we don't have time to buy at a more attractive price.