Hilzoy is mad at conservatives talking about rationing in the public plan. She says that no one's really rationing care with a public plan; anyone can buy what they want. It's just that the public plan will ration for those in its care in order to make coverage affordable.
I feel a little odd putting the shoe on the other foot, here, since this argument is usually used by liberals arguing against libertarians, but surely the point of worry is that many millions of people will be forced into the public system, because its existence will encourage their employers to dump their health care plans. Since private systems have so far found it virtually impossible to deny many treatments for long, this will mean that millions of budget constrained people will find themselves with less available treatment than before.
(I say this assuming arguendo that we think a public plan can and will control costs by limiting treatment--a thesis of which I am actually pretty skeptical.)
This is not a crazy worry. What America is best at is delivering a lot of complicated care in extremis, and "quality of life" treatments. What European countries are best at is delivering a lot of ordinary care for the sorts of things that afflict people from 0-50, which is why most of the Europhile journalists writing about Europe genuinely have very good experiences to report. I'd rather be here to have a hip replacement, but I might rather be in the Netherlands to have a baby. Doing something moderately ordinary here is a hassle. Doing something extraordinary there is often not possible for the overwhelming majority of citizens, though that depends on what, and in what system.