The VA is just the tip of the iceberg. Yes, it is a very large and impactful department, but other agencies suffer from the same problems, corruption and inefficiencies that plague the VA. You are right about the upside-down system of incentives, but another problem is that, in general, there is a total absence of accountability in the government. This culture needs to dramatically change.
The entire government needs a massive overhaul IMO. I’ve worked inside a number of agencies and you always see the same systemic problems. There is a lot of room for improvement. Just saying.
Computer science at the VA is basically in a state of anarchy and many of the jobs the call center people are asked to do are basically impossible. It would have to be less espensive and much more efficient to simply provide veterans with cards (picture IDs of course) which allowed them to seek treatment for pretty much anything at their own discretion.