It sounds like you are proclaiming a need to build emergency funds first, and sure you can do whatever you like, but I think that is a dumb way to go about it. From my point of view it is best to get started investing into bitcoin as soon as possible, and one of the only prerequisites before investing into bitcoin is to make sure that you have discretionary income, whether it is $10 or some other higher amount..and then work on investing every week and figure out your other various bitcoin-related and cashflow management skills along the way while you are already investing into bitcoin within your discretionary income.. with hopefully a goal of investing into bitcoin for 4-10 years or longer. If you cannot invest into bitcoin for 4-10 years or longer, then you might not have enough discretionary income in order to establish that and surely there should be efforts to figure out those matters, yet you might not even have to commit to investing into bitcoin for 4-10 years or longer to get started investing into bitcoin, yet it is still important to consider that 4-10 years or longer is likely the investment timeline for bitcoin.
In as much as I know that emergency funds are very important for someone that wants to start building a portfolio in Bitcoin investment, I also know that there's no where it was written that investors must have emergency funds ready before they can actually invest in Bitcoin, I think people misundertand this things that's why they get confused on how to start.
Just as you said, what every investment need from the start is a constant income no matter how small it is just plan with it and start investing with the amount that can not put on pressure, after all it is just for a start, it doesn't mean that the amount you started with will be maintained like that, one can actually increase his or her DCA once he or she has what it takes to do so, the major thing is to start the journey first because there's danger in waiting and procrastinating with this thing called emergency fund.
For me Bitcoin investment is about consistency and your ability to remain focused and persevere, the amount you use from the beginning doesn't really count rather what matters is your zeal to buy with higher amounts once you get better source of income and hodl for a long-term of about 10years upward.