Freedom of choice allows consumers to select the services and packages which would most benefit them, and single payer healthcare does not allow this.
Customer feedback (through selection of providers of goods and services) is critical, as it allows bad firms to fail, and good ones to continue. The basic income does not interfere with this type of signal, but the single payer system completely destroys it.
The problem is that medical problems are financially bimodal: they tend to be pretty cheap (take some antibiotics) or incredibly expensive (you need dialysis or you have a brain tumour).
So unless you have some sort of useful insurance plan that covers people and minor dependents for less than what you're getting in basic income you'll end up with the problem of people who are unable to get serious medical conditions treated. And making medical treatments universally accessible is part of the point of most single-payer healthcare systems.
The vast majority of consumers are incapable of making those determinations. The few exceptional people that take the lead in their own diagnosis and treatment and become more informed than the medical professionals they interact with are dwarfed by the vastly greater number of people that just want to be told which pills to take at what intervals. There's a massive Dunning-Krueger effect when it comes to self-administration of healthcare. I've met a lot of people who proclaim themselves wiser than doctors but who, when pressed, are unable to properly articulate even the symptomology of their condition.
Customer feedback (through selection of providers of goods and services) is critical, as it allows bad firms to fail, and good ones to continue. The basic income does not interfere with this type of signal, but the single payer system completely destroys it.
Incompetent or unethical mdeical professionals often cling to their positions for years in the private sector, just like they sometimes do in the public sector. Likewise, the public sector is capable of firing or barring such from practice. If you ever live in a country with socialized healthcare for any length of time, then you'll also notice that hospitals and clinics get shut down for lack of efficiency, eg tiny rural hospitals left behind by demographic shifts.
Fully free markets depend on participants being fully informed about market product and services, and upon the ability to defer, invest, or divest whenever it is most economically advantageous to do so. Neither of these conditions obtain where healthcare is concerned. The first I have mentioned above; as for the second, you can manage your risk to some degree, but your ability to predict your medical care requirements is pretty poor - strokes, heart attacks,a nd other less common medical conditions often strike without warning and while they can be insured against (in similar fashion to a derivative) most people are no better at being actuaries than they are being doctors. Don't even get me started on people's inability to choose their own genetic makeup.
I think it's economically inefficient to expect people to develop the huge amount of expertise they need to be better predictors of their own healthcare needs than a doctor or insurer of median competence.
As it includes Medicaid spending as federal benefit spending (and likely PPACA subsidies -- but I haven't gotten that far in the footnotes) the $10k would seemingly amount to just covering every American household's private-market insurance premiums (with no regard to whether they needed them covered and further handing 'free money' to healthy young childless households whose premiums would be much lower).
And without being structured as Medicaid, it would abandon the other (cost-controlling/bureaucracy-reducing/waste-reducing) financial benefits of a single payer health system -- benefits over the private market that Medicaid actually delivers (according to numerous studies).
So the end result is that poor people would see their net benefits ruthlessly slashed, (good luck buying food and shelter after you use that 10k for market-rate insurance) all so that middle-class people can receive a stipend that they don't need and a decrease in federal-dollar-per-hypothetical-unit-of-healthcare efficiency.
The most-charitable read of this proposal, is that it's the result of a naive person doing "black box math" where they seek some mathematical ideal in the re-arrangement of government payments -- without any regard to the benefits being provided, the cost-effectiveness of those benefits and whether the rearrangement is a practical improvement.
I mean, if we're spending $20k per poor person, and children are still going hungry, under what logic would redistributing it as 10k to every single person see any sort of improvement in the ultimate goals of things like keeping children from going hungry?
To not see the massive benefit cut to these people that we've currently defined as needing help, we'd have to assume that for every dollar in tangible benefits received by the poor, there's a dollar in bureaucratic overhead.
And why on earth would we accept such an implicit assumption, when the ultimate source of all these numbers -- the CBO -- has regularly assessed such administrative costs and never finds such waste? [1]
[1] e.g. SNAP, pointed to regularly in the CATO document source, has administrative overhead of less than 4.5% of total federal cost. So if the average SNAP-receiving household is getting $287/mo, the overhead is $13/mo. Under this proposal, food money going to the currently-snap-receiving household would drop to $143/mo, with $15/mo going to 9 households that don't need SNAP. In the name of "saving" that $13/mo. Most of us would look at the infeasibility of a household feeding itself for $143/mo and see their own $15/mo as a more egregious waste than the original $13. Which, cynically speaking, is likely to be a feature. Following such a programs initiation, it would be trivial to get voters behind the proposal to cut their own taxes by $15/mo, to save them the overhead of having the government take the money only to give it back as a stipend they don't need. The end result would be the massive benefit cut and the reduced taxes.
http://www.fns.usda.gov/pd/supplemental-nutrition-assistance...
(EDIT: fleshed out SNAP example)
Personally, the most compelling reasons for Basic Income are getting rid of incentives not to work (since you don't lose any benefits when you get a job) and getting rid of "wrong" ways to be poor (two people with the same income and need can get vastly different benefits simply because of why they're poor).
However, I have zero quantitative analysis to back up the worth of those things, it's just what look good to me.
Sure, at some point, a program can become so complex that the desire to do a greenfield rewrite is very strong. But Second System Syndrome is a real thing. And people taking advantage of a rewrite to move goal-posts and shift focus for personal political/pet-goal reasons, to the detriment of the new system to achieve the existing goals, is a real thing.
I would very much like to refactor the every living hell out of government assistance programs. But I'm extremely wary of people who want to do a fundamental rewrite. Particularly when those people have no historical interest in the program's success. Doubly so, when those people demonstrate they have put no priority on representing, understanding or achieving the original system's goals.