>You don't see the problem here? How do we, as taxpayers, know the money is even being spent for real charitable purposes?

While you are unable to audit any given donor, you are able to audit most DAF program providers by reading their 990 tax form. Here is an example for Charles Schwab, mentioned in the article. https://www.guidestar.org/profile/31-1640316

Even if the 990 is not publicly published, it is required by the IRS and can be audited.

>See for example this story: https://www.telegraph.co.uk/news/2018/06/14/new-york-attorne....

This is less likely to happen with a DAF program because they are managed by 3rd parties with reputations to protect. A properly ran DAF program will research and vet all grantee organizations prior to making a gift, and can (and will) reject donor grant requests if they are found to violate the law/DAF mission.