I totally agree, many people don't get the concept of discretionary income and emergency funds when it comes to using them to invest in bitcoin. Discretionary income is what keeps you from running out of money while you continue stacking the sats, leaving your daily needs no longer secure, but it's not what saves you when life presents you with unexpected expenses, as you rightly mentioned. This is where emergency funds will step in. If you don't, then there's a high risk that when times are bad, you'll sell your Bitcoin holdings and sell too quickly, frequently selling at a bad time.
There must indeed be very clear designated buckets for these funds: first, your living expenses, then build up a safety net of at least 3-6 months of expenses in liquid form, then only use your discretionary income and put it in bitcoin. In this way, you can endure volatility while still having the peace of mind in knowing that you are financially secure. Apart from smart investment, it is also about the sustainability in the long term. Too many investors fail to take this balance into consideration and end up regretting having to dump their coins too early.
Price of Bitcoin is now 115k$ and it's very difficult for most of us to buy one complete Bitcoin at a time. The best way for us right now is to keep investing some fixed amount weekly or monthly or follow DCA strategy. It's also important that during our course of bitcoin accumulation we have necessary funds to meet our daily needs along with emergency needs. That's where discretionary income and emergency funds comes in handy as they keep our Bitcoins intact for longer duration. If you don't have necessary funds to meet daily and emergency needs then your investment strategy is incomplete.