The VA has many problems, but is also wonderful in some areas.
Their care for physical injuries and their EMR are both good. Their EMR, in particular, is fantastic, especially compared to much of the expensive commercial crap that most hospitals use.
Their care for mental injuries and their scheduling and administration are pitiful. Some of that is because the government refuses to fund them enough, but there are plenty of other issues there.
But none of the issues are exclusive to government-run health care. Any place where you have more people who need care than resources to care for them, whether the resources come from the government, industry, or charities. Blaming it entirely on the government just encourages solutions worse than the initial problems.