Hi O-10. Well, I am sorry that I took a jest so seriously, then, though I confess I still am not sure what you were referring to. To answer your question on pit musicians being unbelievable: really the only thing very different about some of them is that some of them, usually woodwind players, are performing on three or even more instruments. These players are called doublers, and they often specialize in that - they are just about all well-trained musicians who often play in free-lance orchestras on the side on whichever is their main instrument. Frogman can speak to that even better than I, as he has actually done some of that. Usually brass players don't tend to do that, though sometimes you might see a guy in a small jazz band playing on both trumpet and trombone.
Otherwise, assuming you are speaking of Broadway show type pit musicians (technically opera and ballet orchestra members are also pit musicians - I have performed all of the above myself many times), this is actually almost the complete opposite of what a jazz musician's performance is. The Broadway shows are very well rehearsed, and then played EXACTLY the same way night after night after night after night after night, etc., never changing - every solo is played exactly the same way every night, or complaints are made about it!! The musicians have absolutely zero flexibility on interpretation. Phantom is going to sound like Phantom every damn time, just like Budweiser tastes like Budweiser every damn time. This is the main reason it sounds so polished. I, for one, could never handle doing that night in and night out. When I was free-lancing in the Bay area, I was one of the first call subs for Phantom, and the most I ever did at any one stretch was two weeks straight one time. I almost went crazy. Now I have also done opera tours that lasted for six to eight weeks, six shows a week, but luckily for me both times it was a Mozart opera that one could never get tired of, and they were double and triple cast, so there was some variety and flexibility in the performances. In a Broadway production, the subs have to come in and play everything exactly the same way as the regulars do. Often you are required to come sit in the pit for a couple of shows to observe before you actually get to sub just for this reason.
I confess that I am not sure why you are blown away that something written out could sound improvised, though - that is not a hard thing to achieve at all. Just about any symphonic pops show is full of many such examples - I have played such written out horn solos myself before in performance. And usually when a symphony does a big band show or something similar, a lead trumpet player is brought in to do the solos, and the rest of us take our stylistic cues from him. All professional musicians have very good ears for these kinds of things and will pick up on how to play in the right style almost immediately. This is not to say that they will suddenly sound like the Ellington or Basie bands, of course, but you get the point.
Also you must remember that anything being improvised on the spot in a jazz club cannot be TOO complicated, otherwise it almost certainly wouldn't work, unless it was perhaps done by a group that was used to playing together all the time, and knew each other's musical tendencies very very well. And of course, as we have talked about before on this thread, the players all know the tune and the chord changes, so the melodic improvisation is taking place inside a very structured framework that all of the members of the group understand. And even if they don't know the tune at all, if they have what they call a "fake book" that includes it, they can use it and get through the tune.