Possibly because that would be illegal (according to the quote you mention) and so they would likely lose further negotiating ground by doing so.
Then again, there was this Reagan fire all traffic controllers and put them on a black list thing in the eighties https://www.history.com/this-day-in-history/reagan-fires-113... but I have a suspicion private company with no military personnel to immediately take over would be more willing to work something out.