The sad part is that in Star Fox 64 3D, he did have his classic, cocky attitude.
For some reason, SSB4 dropped the ball with voice acting. There's recycled voice clips from previous games with Roger Craig Smith's Sonic for Sonic, Mike West doesn't sound annoying in Star Fox 64 and 64 3D, but does in SSB4 - whenever Fox talks... -, Mark Lund sounds like Sly Cooper - voiced by Kevin Miller - when he says Falco's Final Smash quote and he sounds bored, and then there's DK, Diddy, and Bowser sounding like an actual gorilla and monkey and a monster instead of the goofy voices they have now. It's much weirder when you hear Bowser Jr. sounding cute and cartoony next to his dad, Bowser.
As a character, he acts as a foil to Fox's calmness similar to how Mario's the stoic hero and Luigi's the wacky fellow or how Marth's graceful while Ike's brash and brutal. It gives charm to the cast, but here's the thing, Star Fox has Star Wolf acting as their foil to a good group of mercenaries and an originally bad, but now more chaotic neutral band of mercenaries and space pirates. So, it's like an option of do we want the rival or the wingman?
Kirby has both the rival and the "villain" through Meta Knight and King Dedede who have different ways of fighting. Falco and Wolf were introduced as clones where Wolf was originally intended to be in Melee, but scrapped because he would look too similar to Fox (despite having an eyepatch, different jacket in the Star Fox art, and probably being larger than Fox). They're like Ryu, Ken, and Akuma, except Wolf became more like Gouken or Iori Yagami to Fox's Kyo Kusanagi. Still, they were conceptualized as fighters being similar to Fox and not having much to be different despite Assault introducing a ton of weapons and Falco could have had an entirely different moveset. Oh well, that's them as fighters.
As a character, it would be nice to have Falco and Wolf and maybe even Krystal, Leon, Panther, Peppy, and Slippy. It's just that Falco and Wolf seem to have a larger impact on Fox than the others with Falco being Fox's wingman, a trusted friend, and the ace pilot (despite every game making Falco look like a fool) and Wolf as Fox's rival with a rough past - it's implied Wolf might have been involved with James McCloud's death.
Honestly, Star Fox could have had only one representer or even be scrapped entirely after Melee because since Assault, Star Fox wasn't that lively. Now we have a remake to Star Fox 64 and Star Fox Wii U close to release which might help boost more awareness and life into the series. Fox, Falco, and Wolf are still more known through SSB than their home series like Captain Falcon and Marth. Samus and Captain Falcon represent their series alone and newcomers will represent their series alone because of their status as newcomers and because some series don't have any notable characters to add. I mean, who are you going to add to Game & Watch and ROB's "series"? Animal Crossing at least has characters who are colorful, but there's not a lot to do for combat and Xenoblade could function like Fire Emblem where new heroes are added.
It would still be disappointing to anyone when characters are cut. It means a character that could have been improved and evolved with each, new game and it means one less character to represent a series, especially if that series has a small amount like Mother/EarthBound or the Ice Climbers. Fox and Falco were a bit like Batman and Robin/Nightwing with Wolf being a bit like... er... I guess Jason Todd as the Red Hood as a rival. Without Falco and Wolf, it just feels incomplete since they were there, but not anymore. There's no longer a wisecracking, cocky bird or the vicious badass; only the hero stays to fight without his friend and rival.