I don't know how other universal health care systems are run, but in Canada you have a fair amount of choice when it comes to who provides your care. The government isn't directly making any medical decisions or directly providing patient care. In other words, it's single payer, but not single provider. The government essentially acts as a single insurance company, where choices about what care is given are left up to the attending physician and are subject to available resources as in any system.
It's not a perfect system, but it's also not what most people think it is.