The one time Captain Falcon used the Falcon Punch outside of Smash was in the final episode of the F-Zero anime. Captain Falcon sacrifices himself to defeat Blood Falcon with one final Falcon Punch.
If you beat Master Mode (and all previous cups before), you're treated to an alternate title screen.
This title screen features the inner parts of the Blue Falcon.
The driver's seat is empty.
A slight interpretation can be gained when you beat all cups prior to starting Master Mode.
Beating the game on normal difficulty ends with Captain Falcon mentioning, "My time has not yet passed."
After beating the game in the hardest difficulty (and Master Mode), Captain Falcon announces he retires as an undefeated champion at the Winner's Circle.
It can be assumed this is why the Blue Falcon is empty after unlocking this final title screen. Rest easy, Falcon.
This game is a treat for long-suffering F-Zero fans. It might not be exactly what some of us hoped for, but admit it, F-Zero lends itself really well to this concept. From what I've played so far, it's actually really well done and a lot of fun! I expect to be playing a lot of this.
I already managed to get 4th place in my first mini prix.