The Pentagon was hit around 9:40, which is about when Flight 93 stopped responding. Nobody has ever suggested that Cheney ordered Flight 93 shot down before that moment. But at that point, by the author's timeline, the Toledo F-16s had already reached NYC. Why on earth would NORAD dispatch F-16s from New York to shoot down a plane near Cleveland, when there were three F-16s already in the air over DC?
If you want to believe in a conspiracy theory, it's much, much simpler to posit that NORAD sent one of the three DC CAP planes north to shoot it down. That theory only requires the DC CAP pilots and their commanding officers to be in on the conspiracy, and is at least plausible. The author's theory drags in everyone from the entire staff of that airfield in Toledo to the air traffic controllers in New York City, way too many people. I'd sooner believe the 9:22 sonic boom was caused by a deorbiting UFO than that the government could keep hundreds of people quiet about shooting down a civilian airliner.
But the key point here, is that your line of reasoning assumes that the Toledo F-16s would only have been ordered to pursue Flight 93 pursuant to a shoot-down order. That assumption is not well founded. For example, all the military jets were scrambled, and told to intercept the hijacked airliners, well before anyone has suggested that any order was given to shoot down any airliners. The Toledo F-16s might have been ordered to pursue Flight 93 before any decision had been made regarding the shoot-down policy.
Something caused that sonic boom. The conceivable choices are:
I'll take what's behind door number 3, Monty.