There was some of that, but perhaps not in the sense you intend...
Eleanor Holmes Norton (even a stopped clock...) is correct in saying that some of the voucher students were not "rescued" from failing public schools...some were already attending the private schools on private scholarships or other funds, and then received public funding to keep going to the private schools.
If you read the studies, they don't seem to have received as many applications as they'd hoped from the very worst schools, and they weren't sure why, except that there was a lot of paperwork & initiative involved to participate in the program - parents had to file (apparently complicated) applications, & research and apply to the private schools seperately. They also had to figure out transportation to & from the private schools, if their children were accepted. Lots of hurdles, especially for minimally educated parents in the inner city.
A great school can’t completely compensate for bad (or non existing) parents I guess.