Back in the real world, I think people might have a better understanding of their finances if they could view their position as a continuous function rather than only seeing the discrete transactions as most of us currently do.
One of the things that distinguishes basic income from welfare is that the former is an unconditional transfer payment. That means that some of the folks who receive it don't actually need it. But it gains a lot of advantages over traditional welfare plans in the process - it is much simpler to administer, it doesn't require making politically fraught judgments of who is worthy, it eliminates disincentives to work that come from income cliffs, and it's easier to gather support since everybody sees a benefit.
Presumably those who don't need the basic income will end up funding a greater portion of it through higher taxes.
How do you 'enforce' competition? Build baby build, and you set how much to build by how much is being charged in an area.
As for what happens when that gets blown out because of an emergency, I think addressing the cost of health care in a place like America is an independent and arguably more important challenge than providing BI. The system built around insurance policies and over-priced hospital procedures is ridiculous to say the least. You shouldn't need insurance to afford simple checkups, prescriptions and treatment, which would prevent most overcomplicated health conditions for the financially constrained.
Every other financial emergency I believe can be responded to with "tough shit you've exhausted your social safety nets," and it's probably ok. BI won't solve everything (or even anything) magically overnight, the intent is to build a culture long-term where people manage their lives around it and don't have to be told "tough shit" from birth.