So perhaps I should have said 'the implementation of immigration is entirely within the control of the executive'. There are extensive rules with discretionary holes large enough to admit or deny the entire population of Afghanistan.

For any single individual, the choice on whether to let them in or not is up to the executive branch. It is far more common for them to use their discretionary options to deny people entry, but this is a magical case where they are actually doing the opposite - "DHS Secretary Alejandro Mayorkas is using his humanitarian parole authority to allow at-risk Afghans without visas to enter the U.S" https://twitter.com/camiloreports/status/1430198432434737153...