Is the underlying problem with the VA hospitals scandals greed?

Perhaps self-interest in a better word to describe what is going here.

There is a wide spread and naive notion that for-profit institutions,  aka business, are driven by greed and that dishonesty and bad motives dominate their existence and that non-profit organizations are the opposite in every regard; But folks who populate non-profit organizations are cut from the same cloth as the rest of humanity and for them as for everyone incentives matter.

This commentary by Glen Reynolds gets it right.

I quote from  his  comments from USA Today:

"In other words, they cooked the books. And what's more, they did it to ensure bigger "performance bonuses." The performance may have been fake, but the bonuses were real. (One whistle-blower compared the operation to a "crime syndicate.")
And that captures an important point. People sometimes think that government or "nonprofit" operations will be run more honestly than for-profit businesses because the businesses operate on the basis of "greed." But, in fact, greed is a human characteristic that is present in any organization made up of humans. It's all about incentives." ....And, ironically, a for-profit medical system might actually offer employees less room for greed than a government system. That's because VA patients were stuck with the VA. If wait times were long, they just had to wait, or do without care. In a free-market system, a provider whose wait times were too long would lose business, and even if the employees faked up the wait-time numbers, that loss of business would show up on the bottom line. That would lead top managers to act, or lose their jobs."

If you look at the history of the VA system you will see greed and corruption boiling over the top at the very beginning..The historian  Burt Folsom gives a brief review of the origin of the VA system and the corruption and mismanagement that characterized its early days under the administration of President  Warren Harding.

The point is that people act in their self interest ( when their actions rub up against our moral priors we call it greed) and that markets impose the discipline of profit and loss that are lacking in  monopolies such as the socialized medicine of the VA system and often -but not always- direct that greed to the benefit of others.

As Milton Friedman said the question is: under what system will
greed lead to the  least harm,his answer was capitalism.Here is his priceless reply to Phil Donahue .