Two things wrong with your comment. First, the VA isn't the military, it's the VA. So what the VA does has nothing to do with what the military thinks of wants to do, because it isn't part of the DoD. Second, the VA is a comparatively good American healthcare system, you've just been lied to, mostly by 1) people who don't like it but have relatively little context, 2) people who want to paint it as irreparably broken so they can privatize it.
There are problems with access that have to do, significantly, with funding. The system has to do its best with what Congress appropriates. In every mental health clinic I've worked in within the VA system, we are stretched, we need more providers that we can't get and the ones who are there work like dogs, and provide care about as efficiently as they can. Throughout all that stretch, a Veteran who calls up my clinic can get mental health services initiated, generally, within the week. (If it's an emergency, they can show up during business hours and talk to a provider within the hour, often within ten minutes.) Try getting either in the private sector.
Try finding out about difficulties in access in the private sector, try finding out about wait times, try finding out about sterilization compliance, try finding out about what they do to ensure high levels of training and adherence to the best empirically-supported mental healthcare interventions systemwide. Good luck: they're not answerable to you to Congress, to FOIA requests. The appropriate control group for VA services is the private sector. To the extent that we're able to make any head to head comparisons, we generally do "equal or better." But that ability is limited, because you simply won't be able to find out as much bad stuff about them, as they lack the transparency of a public system.
http://www.gallup.com/poll/186527/americans-government-health-plans-satisfied.aspx