It sounds like a problem I had in my 03 zo6. Normal shifting was excellent, but when you went through the gears quickly it was extremely difficult or took a while to get into the next gear. its a common factory flaw and the problem lies with the line from the clutch master to the clutch slave cylinder and it being too small of a diameter to transfer fluid quickly. Basically creating an critical orifice that wont allow the fluid to pass through as quickly as you can press the pedal.
Do a Google search for "LS1 drill mod" or "t56 drill mod". you'll get alot of hits and DIY's.
Your clutch disk isnt the problem, but with the power of the LS motor, a sure fire way to tell if its too thin is if it slips under load. like getting on the freeway, sticking it in 3rd or 4th, and gassing it, if the RPM goes up without an equal amount of acceleration in the car, then the clutch is slipping. that's usually the main indication of a worn out clutch. What you see under the car when your daughter presses the pedal sounds pretty normal.
Lastly, the restriction in the clutch line works both ways, restricting it when pressing the pedal. As well as when releasing the pedal, that could be why the people at the track said they smelled your clutch burning. It probably wasn't fully engaged during acceleration.
-Dan