The fact is, they certainly can, and I suspect they would in order to keep the largest carrier of US domestic passengers flying.
You're still missing the point. The goal isn't to strike, the goal is to get a good deal. A PEB can be a way to achieve that. The deal struck after the PEB in '97 at American is a perfect example.
Strikes are rare and PEB's are much rarer, especially in the airline industry. My question is this, suspend your disbelief for a moment and speculate that the NMB has released SWA to a 30 day cooling off period. Do you agree that the NMB would be required to notify the president of a situation that "threatens substantially to interrupt interstate commerce in any section of the nation" and that he or she would convene a PEB after that recommendation? If so, what do you think the odds of a strike actually taking place?
You still don't understand the process. Before the NMB even issues a release they undergo an impact study to ascertain what the impact would be to the economy and the transportation system. That's just part of their due diligence. The NMB is also in constant contact with the White House, so it's not something that they just suddenly call the President about during a cooling off period. The President's chief counsel or Chief of Staff would have alerted him to a cooling off period long before it was actually issued. But the White House's goal is the same as the NMB's: get a deal reached without a strike. And this is the part you need to remember: sometimes a release is the only way to reach a consensual deal. That is the NMB's calculation. The NMB will release you into a cooling off period when they believe that the release is likely to lead to a deal so that the labor unrest can be cleared off the government's table. A release is never issued if the NMB (or the White House) believes that a strike is the likely result of the release. The release is there to apply pressure to both sides to reach a deal.