Lack of oversight, increased number of nonprofit groups complicate giving to veterans' charities...

Agree. You really have to do your homework before making any donation. It's not just nonprofits that claim to help veterans, its all of them. Need to do a search online for a nonprofit before making any contribution. I'm not going to recommend any one site over another but there are many out there. Just about every TV station in Florida is currently posting links to these sites because of the fraud around organizations soliciting funds for Orlando shooting victims. Key factors to look for are size of accounts that you mention, payrolls of executives and percentage of giving vs overhead.

In some cases it is a failure to enact tough state laws regarding charitable organizations, and in almost every state a complete failure to provide appropriate oversight. Unfortunately, in a few cases, it is criminal intent to circumvent laws for self-enrichment and/or embezzlement. In the case of private foundations it is often the desire of the founders to divert money to relatives for supposed management in lieu of actually doing anything to promote the common good.