No!!! The superchargers don't use the car's internal charger, they charge the battery directly.
The cars own charger is used when you're on AC power, such as at home or at some destination (hotel) which has a tesla wall charger (HPWC).
In that case the standard charger can suck 11kW max, but the upgraded charger can draw 16.5kW.
So basically, the only reason to pay for the upgraded charger is if you really need to charge fast at home (if you have a wall charger installed) or at some destination.
Usually destinations are overnight (hotel), so even the standard charger has plenty of time to fully recharge the car overnight (takes 6-8 hrs from empty, depending on battery size).
The upgraded charger would be useful if you were going to stop at a destination charger for lunch to grab some extra juice for a long trip. That's about it.