I'm a vet that used VA health care as my primary means of health care for a couple of years, so I'll try to give some clarification. And I apologize ahead of time for the egregious use of acronyms.
VA health care isn't necessarily free. When you first apply for it, you get assigned to different priority groups based on service-connected injuries / disabilities, income and whether or not you're a combat vet. Most priority groups lead to a vet being able to get free health care within the VA system, but a few of the lower priority groups lead to vets having to pay copayments.
The main issue with the VA is the number of facilities (that can actually care for someone) and wait times. So there's like 3 level of health care facility in the VA. You have CBOCs (Community-Based Outpatient Clinics) at the lowest level, OPC (Outpatient Clinics) in the middle and VAMC (VA Medical Centers) at the top. CBOCs are literally just clinics where you can go see a doctor if you're in pain, sick or for like a yearly physical. OPCs are more like a normal hospital and they can do stuff like X-rays or more involved testing, plus they have some specialists in-house, but they don't have emergency rooms or urgent care (at least none of the ones around me do). VAMCs are like medical complexes. These are where the VA emergency rooms are and these are the places where stuff like surgeries happen.
There are more CBOCs than OPCs or VAMCs. There are more OPCs than VAMCs and most states only have like 1 or 2 VAMCs. This means that if you have an emergency and you want to make use of that free healthcare by going to a VA emergency room, you have to go to a VAMC, which for a lot of people will be too far away. This leads to people going to local hospital emergency rooms. The VA is supposed to cover the costs of a visit to a local emergency room, so long as you meet their criteria, but even if you do meet the criteria, they will more often than not try to pin you with the bill.
Doctors in the CBOCs can only really refer you to an OPC or VAMC if you have something requiring X-rays, specialists or anything more involved than checking your ears or drawing blood. I used to go to a CBOC and the first time I met my doctor, the first thing he told me was that if I ever had a problem that seemed like it would require an X-ray, don't even bother coming to see him because there was nothing he could do but refer me to another place. But anyways, if you get referred to an OPC or VAMC from a doctor at a CBOC, you have to wait like 3 days - 2 weeks in order for a scheduler to get in touch with you to schedule an appointment. Then, your appointment is very likely going to be another 2 weeks - 6 months away, which is again going to likely cause people to roll the dice and try their luck at a local hospital where the VA, again, will likely try to pin the vet with the bill.
Earlier this year, I got pinned with a bill for just under $5,000 for a visit to a local hospital's ER because I couldn't drive the 2 hours to the closest VAMC to go to their emergency room. And a kind of funny, but rage inducing conclusion just happened with that bill a couple of weeks ago. I got a followup letter from the VA saying that they paid a $5 administration fee to the hospital on my behalf in regards to that bill. All I could do was shake my head.