Except it is possible.. What you mean is that the people who pay you don't want to pay you in bitcoin, and the places you buy from aren't taking bitcoin. But they could if they wanted to. It is in fact possible today.
You asked "how does being your own bank do these things". I think its pretty clear how they do those things and how banks are not necessary for you to pay someone else (in bitcoin) today.
> would get into a lot of legal trouble if they didn't pay payroll taxes
Paying employees in bitcoin does not prevent them from paying payroll taxes...
> huge chicken-and-egg problem
Its really not tho. Individual merchants can (and do) accept bitcoin without all their customers willing or able to pay in bitcoin. Its just another option. Individual employers can offer the option of paying employees in bitcion (especially easy if they're a merchant that accepts bitcoin). Individual employees can opt into such an option. There is no chicken and egg problem, it can happen gradually one person at a time and one company at a time.
My point is that none of this is actually possible. My employer doesn't pay me in Bitcoin/etc, has no facility to right now, and would get into a lot of legal trouble if they didn't pay payroll taxes. I don't frequent any shops or online services that accept payment with cryptocurrencies. My landlord wants to be paid in GBP because that's what he's paying his mortgage in and he doesn't want the hassle and cost of conversion.
I get that these things are all possible when everyone has moved over to cryptocurrencies, but that's a huge chicken-and-egg problem, and I've yet to see a path from where we are now to this future world that is at all credible/accessible/legal/compliant/safe/cheap.