Gradius doesn't have characters with personalities talking to each other. Gradius doesn't focus on actually telling a story with character development and relationships in the heat of combat. Gradius doesn't present a real world with characters you know getting hurt or dying. Star Fox does.
This is why Star Fox can evolve into something more. When it evolves can developers ignore the aspects that make the game great? No. They need shoot'em up action. They need campy and fun dialog. But more could be done with it. In fact, it could be made even better...and still keep that shoot'em up feel.
Imagine. a ground level on foot. Game play is based on the top down Smash TV type run and gun. Now the Wii U tablet shows where your allies are, and in real time displays how they are fighting and doing. Are they in trouble or what not? Falco rushes in head strong and is doing his part well...an is calling out snarky remarks about how many he is taking out, and that the team is taking too long. You are fighting waves of enemies, but then you hear slippy is in trouble. You look down at his map and see him pinned down. You can command him to make his way towards you, or you can move to him. You notice the enemies are mostly between you and him, and he can not move towards you safely, so you run to him. Again, the game play is pure action.
Fox can even have a dodge that activates like a barrel roll to get out of danger. This type of game play that is not a first person or 3rd person traditional shooter...would add variety and I think fun to the game...if it is designed well.