They'd probably be perfectly happy to see it turned over to and managed by private equity parasites. NTSB is a different animal - though not part of DOT, NTSB used to report to it for administrative purposes, while at the same time doing investigations into the FAA, which was also under DOT. Because of that conflict, Congress made NTSB fully independent. The problem with taking SSA out of the executive branch is that it's tied up with the IRS, which collects the FICA tax that funds it.
Also there's the unfortunate Flemming v. Nestor case from 1960, which addressed an amendment to the Social Security Act that stripped benefits from contributors who were deported under the Immigration and Nationality Act. A resident alien from Bulgaria who had paid into Social Security for 19 years began drawing benefits. He was subsequently deported for involvement in the Communist Party, and his benefits were terminated. He claimed that the amendment had deprived him of a property interest in Social Security without due process and was therefore invalid. SCOTUS ruled that there is no contractual right to receive SS payments because they are not “property” and are not protected by the takings clause of the 5th Amendment.
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