I haven't played any of the games that came after Star Fox 64, but from what I've heard, ever since then the series started putting more and more emphasis on story and interpersonal relationships between the members of Star Fox (to the point that Command was almost like a bloody soap opera), and less on shooting aliens and space monkey faces. That shit obviously needs to change.
As for how to make it more than a run-of-the-mill rail shooter, I think they should take a cue from Star Fox 2.
(I am retarded and don't know how to embed)
For those of you not in the know, Star Fox 2 operates very differently from the original and Star Fox 64. The game runs in semi-real time, so the game remains paused when you're not doing anything, but when you move or perform any action, the clock starts ticking and all the enemies begin moving (kind of like in Mount & Blade, if that helps anyone at all). The primary objective is to liberate the planets in the Lylat System while simultaneously preventing intercepting missiles and enemy battleships before they can reach Corneria. During enemy encounters and planet missions, the clock slows down, but doesn't stop, so if you plan it poorly or just fuck up, you might have to interrupt the mission in order to intercept an enemy. To keep flying around the map from becoming tedious and to actually make the rest of the Star Fox team do something useful, you can control multiple characters, each with their own statistics and unique starfighters. (So Falco could have improved maneuverability, Peppy could have advanced barrel roll and bomb-using techniques, and I guess Slippy could be like a joke character since nobody would want to play as Slippy anyway)
To add further replay value, the enemy movements and spawn locations could be randomized, liberated planets could be recaptured by the enemy if left unchecked for too long, and if you were to ignore a planet for too long, that part could become infested with enemies, and become harder to complete. Sort of like the Panic mechanic in XCOM: Enemy Unknown.
It would give you an idea of a much larger conflict than a standard linear rail shooter would, so you'd really feel like you're defending an entire solar system. Don't tell me that wouldn't be amazing, you lying fuck, you.