Loved or hated, the VA is a large HMO set up as a benefit for veterans to help them because many of them have had to sacrifice their personal health to fulfill orders their were given.
Any healthcare system is riddle with anecdotes of failures, but if you look at it from a population health perspective, as a whole the VA has been able to care for its patients with a pretty good quality if you consider the cost constraints they are working with. I'm not aware of any private health system that's as efficient as the VA.
Also, the VA has had a fair amount of innovation that then trickles down to the rest of the healthcare systems. One example is BlueButton, a method to obtain a dump of all your healthcare records as a patient. Eventually CMS (Medicare) enhanced it and started promoting for everyone else, under the spec "BlueButton Plus".
On that topic, the US as a whole ranks 24th in the world in terms of preventable deaths[1], and we spend twice per capita as the highest-performing country on the list. Our healthcare has A LONG WAYS TO GO to improve.
Back to the original topic, I'm not speaking from the POV of someone having experienced the VA care system. I'm speaking from having worked in population health. Quality is defined by specific measures that compare in aggregate the outcomes of patients given particular co-morbidities. That doesn't make it perfect, it just means that as a whole, the VA treats people as good or better than the private system.
As for wait times to see providers, keep in mind that many people on the private system may be able to see specialists but just can't afford it, so they never get treated... until it's too late. Part of the quality-based initiatives are to nudge providers (via reimbursements/penalties) to treat things while it's early enough, both for the health of patients and cost reduction—instead of waiting for complications.
[1] http://www.oecd.org/officialdocuments/publicdisplaydocumentp...
Outcomes do take into account the starting conditions ("comorbidities") so this doesn't necessarily mean that the actual health of that population is better than average population. Because as you pointed out, service people get a pretty raw deal when it comes to occupational hazard. ;(
Additional info:
RAND analysis about VA and non-VA care -- http://www.rand.org/blog/2012/08/socialized-or-not-we-can-le...
Meta-study commissioned by the VA looking at existing studies comparing quality of care in VA and out of VA -- http://www.hsrd.research.va.gov/publications/esp/quality.pdf
As this point the US should give the veterans the same insurance place Congress gets.
Most of us use our own health insurance from our jobs because the VA is absolute garbage.