In response to the 9/11 attacks 2 F-16s were launched from Langley AFB, in southern VA. at 9:22. This was too late to reach DC before the Pentagon crash and for reasons that are not really clear they were initially sent out over the Atlantic. Also Bush did not issued his shoot-down order till after the Pentagon had been hit.
But let’s imagine history had happened a little bit differently: the fighters took off a few minutes earlier, headed directly for DC and got there before crash at which point the POTUS had given the order. Would it have made sense to have shot down the 757? Eleven people were killed on the ground in Lockerbie after a bomb exploded in a 747 flying over it. A 747 is about 3x more massive than a 757 and flight 103 had a lot more fuel than flight 77 on the other hand flight 103 was at about 11,000 feet leading the debris to be dispersed over a wide area and Alexandria, especially the area around the Pentagon is very densely populated. According to Wikipedia the population density was 8,452 or- 9,493 in 2010 -11 but it would have been higher along the plane’s approach path on 9/11. I was not able to info on the Scottish town’s size or density but its total population is only about 4000. There was a traffic jam on the portion of highway the plane flew over and there was only about 250 of lawn between the Columbia Pike and the Pentagon. Since the 757 was flying at about 750 feet/sec. I doubt a fighter pilot could have timed the shot to have hit at the precise moment. To make a long story short I imagine a lot more than 11 people would have been killed if 77 had been shot down, perhaps more than were killed in the Pentagon. And not to sound insensitive but while there were many civilians in the building most occupants were military personnel, a career in which risking your life to protect civilians is part of the job description. Would it have made sense to sacrifice the latter to protect the former?