Not true. My carrier has employed at least 2 non-us citizens, one of whom was hired after 9/11. I know of other non-us citizens hires by US part 121 carriers on the post 9/11 era.
A whole complete two out of how many on your seniority list (including the unworthy and miserable miscreants who fail to do their Jepp revisions and wear a hat) since 9/11/01?
I'll bet it's not exactly a flood through an open conduit and that the two ( plus a possible bit) out of hundreds if not thousands must have held a green card otherwise their employment would be illegal.