The problem with this thinking is the "would have" and "could have" are vastly different than "did".
In reality he purchased the guns from dealers and actually passed a background check that he should not have. I think this makes the Air Force liable.
No, people who want guns who know that they won't pass a background check just get them from a source that doesn't require a background check. In the state of Texas this simply means buying them from any other citizen who is not a gun dealer or having a friend or family member buy them for you. This is why all this crowing about "background checks" for gun ownership is so ridiculous.
The Air Force is no more responsible for this tragedy than if you sold your car to someone who had a history of drunk driving or sold your old pressure cooker at a garage sale to a Muslim illegal immigrant.
The reason the Air Force might settle if they are sued is because juries in this country like to give deep pocket money to victims regardless of who is to blame and what the law actually says.