With PreScreen Videos under way now, there are plenty of opportunities to redo videos that have a flaw in them. Maybe you end up with lots of videos with different flaws in them. Shouldn’t you theoretically be able to redo them until you get one just right? Or, maybe at some point, you just say “this one is good enough”. Where is the threshold between not trying hard enough to get it right and just saying “well, there you go, this is my playing level, hope you like it.”
Think authenticity not perfection.
If it doesn’t truly represent your son bc he’s obviously nervous or just having a bad take, re-do it. But if each take has a little error here or there, that’s him…still young and ready to learn. Someone might hear the error and think …ah I can help him with that. I’m quite certain my Ds prescreening were far from perfect but they sounded like her. So that was what we submitted. At some point hopefully your son will be able to chose the one(s) he likes best and represent himself honestly and confidently.
I second @bridgenail … as long as the recordings represent your S’s playing accurately then it is fine.
During S’s recording session, if a take was pretty good but didn’t accurately display his strengths, for instance if intonation is generally excellent but there was an uncharacteristic off note, he would ask for another take. We had time constraints with the paid musicians and studio, so couldn’t do more than a couple takes of each tune, and a couple were one-and-done. During the session, I wished we had more time, as I didn’t feel S was playing at his very best, (he was recovering from the flu), but in retrospect, it was fine. I think the people looking at the pre-screens know pretty quickly if they are seeing what they want for a second look/audition.
Best of luck to you and your S with your recordings! 
For me, that was the most anxiety inducing part of the year. It was the culmination of all the researching and planning and helping to prepare. Once the prescreens are recorded and sent, things are set in motion and you just try to relax and roll with it as the results come in. Exciting times!
@indeestudios - my anxiety occurs earlier in the process, and it has already passed. I worry a lot up front, and almost feel like something is old and done by the time it happens. Just a personality note (INTJ would you guess?)
The one flaw that stands out to me (and I see fewer than S does) is a bow squeak on In A Sentimental Mood. Otherwise it is a pretty moving song. It was 3 takes on that one.
These particular songs were done yesterday at home with play-along (something called I Real Pro, I believe. Very handy program can be programmed for duration and such). Today we do some live recordings and have only so many takes with those. I notice S is getting better at realizing how to meet the most requirements in the fewest songs and other basic logistics and resource management concepts - I had been hoping those would develop.
Jazz pre-screens can be tough if your son is like mine and every single take sounds completely different. I don’t don’t that my son could repeat a solo if his life depended on it. In the end, he chose the accurate head and good vocabulary over perfect playing (i.e. squeaks and such.)
His HS combo cut a record in the spring and did the recording over two sessions. It made a big difference for them to be able to listen to their first session takes and then be able to back in the studio. I think that’s the ideal situation.
Keep in mind that the bar on pre screens is not as high as the audition, it is a weeding out, and IME/IMO it is unlikely that simple mistakes like a squeak or a slip would keep you out, from what I have been told it is things like bad intonation (on a string instrument) on a continuous basis, rather than a lapse, that would keep you out. More importantly, even a professional doing that audition material would likely have slips and slops, it happens, and a pre screen that is too perfect might get them suspicious. Others have said be yourself on these threads, and I agree, and unless you feel a take is absolutely horrible, had too many mistakes that were basic (like poor intonation all the way through, or playing something rhythmatically wrong, etc), if it sounds decent it is probably okay to use.
Very timely thread as we are recording etudes this week. Thanks for the feedback, veteran music parents! It is helpful to all of us going through this process for the first time.