There are just two big problems with these assertions. The first is that presidential impoundment dates back to the very beginnings of the nation. The second is that letting presidents impound funds appears to have been an effective tool for keeping federal spending under control.
Impoundment is just a jargony word for instances where Congress appropriates a certain amount of money for a program in a given year, and the president refuses to spend all of it.
The reason they do not want the president impounding is that the pork factor may just get eliminated, so there will be no more bridges to nowhere and other logrolling goodies. It imposes discipline on congress, which has been lacking since Nixon.