It's simple economics: third party payment. Patients want as much care as possible, doctors are incentivised to provide it and insurance companies pay for it. Until that changes healthcare costs will not decrease
Honestly my current theory about health care costs is that they're high - and are inflating - because people like health care. There's a basket of potential goods and people really like that particular item. To get it they spend what they have and then pressure politicians to get them ways to spend money they don't have. They complain more and so the politicians find ways to hide the costs from them.
But just because vet costs are increasing at a similar rate (or more) than human costs doesn't mean the third party vs first party thing isn't the issue. Absolute overall human health care costs are still way higher then vet costs for similar services(1), despite the rates of increase the past few years.
Also, anecdotally, some vets are going the other way. There's a vet near my house, for example, that just started weekend + holiday hours with rates the same as to their normal ones. We hadn't even gone their before, but when my dog woke up last Easter with some eye problem, we called around, found them, and got some ointment that helped. The other two options would have been: wait a day, or go into an emergency clinic that's k's of dollars. If we had pet insurance I'm sure we just would have gone in without thinking.
1: http://parkhillvet.com/2012/09/comparing-costs-of-human-vs-p...
That is the bizarre part of the US system - it might very well work better than today if it was actually a free market, but they've managed to get the worst of both: A highly regulated market where the regulations doesn't ensure proper service.