I wouldn't say that Star Fox needs to reset. I wouldn't say that they need to specifically merge different playstyles precisely how you suggest, either.
What was great about Starfox and StarFox 64 was the impending rush to save everyone, first and foremost. The game begins with an attack on Corneria, the last planet of the Lylat system not at least partially controlled by Andross. StarFox is hired quite literally after the entire galaxy has been taken over by Andross's blitz and overall manpower. You start, with a briefing, and you go straight into action and directly see the true enemy and why he's such a problem. As you go planet to planet, you always arrive arrive Andross has assualted something, and you're not looking for people, information, or anything like that, you're looking to weaken Andross's legion of henchmen, machines, or bioweapons, so you don't get too overwhelmed when you finally get to him.
StarFox heads to Venom destroying everything of Andross's in sight, and soon, when they reach, they destroy one of two of Venom's defense systems by themselves. They are greeted by the anti-heroes, and have to earn their way to Andross. There's little story, just that Fox and Co. save the day, then go on to save the galaxy after it's already too late. Everything is against their side already. When they go in, they move quickly, making a bee-line to the source of the problem.
Where Assualt messes up is the when you, as StarFox are hired to prevent the aparoids from taking over. It's messed up when you have steal information. You are mercenaries, yes, but remember, you're very expensive. You're used at the last minute, at the very last chance, and paid salaries that only entire planets could afford. You don't hire these guys to gather information unless that info is behind about 2000 enemies and you want everyone to know you have it. You don't get them to go and protect something that may not even need defense. It's terrible pacing. StarFox is about style, desperation, and tension, and while Assualt has tension, aside from character design, it lacks style and precludes desperation until the very end of the story.
I haven't played Command outside of multiplayer. It's on my list. However, I don't like the "draw your approach" and very many multiple paths concept, it seems too out there for the linear, planned first two Starfox games.
I didn't ever play Starfox 2, few did, though I watched a speed-run of it once. In it, there was strategy involved, allowing you to choose which stage you wanted to play on in an attempt to lower the defenses against a final destination. Once again, StarFox was called in late, the galaxy was nearly under the enemies controls, and to make up for the non-linearity of stage design, you had limited resources to choose stages with and you carried damage from stage to stage.
So StarFox needs desperation. When you begin the game, you should see a briefing telling you how bad the Cornerians or whomever is in their mess, with a plea begging for help at the end. Always. That should be tradition, because it thrusts you straight into the action, just like how StarFox team would be called in. Next, after you help, you need a reason as to why your contract's enemy is also your enemy. After all, as mercenaries, you'd want to go with the top bidder of the conflict, and if you're strong enough to win against amazing and astounding militias, then there would likely be a bidding war. Give me a reason why side B didn't hire Starfox, and we're fine here. This works out for Assualt, actually, since aparoids are going to destroy everyone.
Now then, StarFox isn't about being complicated. It's about being fast and fun. If StarFox were to branch out beyond the main StarFox series, the games would need to have the same feel as StarFox 64, but they don't have to have the same gameplay. Just keep things fast and desperate the entire time, and the game would have that feel. If you wanted to set up an RTS or Turn-based strategy game, and they don't meet that criteria, then don't call it StarFox. I didn't say it couldn't be on Corneria or based in the Lylat system, but the game wouldn't likely be StarFox.
If you wanted a game that required mercenaries with no cause other than cash, have it star StarWolf and his team. They aren't supposed to be innately bad, they are just the anti-heroes. They can be the ones with the upgrades and all that jazz. They can go seek out secret information by killing everyone on a space station. Whatever, that's not StarFox's typical job.
If you want all these things in a single game, do the Sonic Adventure thing: Divide up the game. Sure, people began to dislike the 3D Sonic games. Why? Partly because they tried to age the series by offering alternative gameplay, such as poor missions where you infiltrate buildings going slowly with mechs, and sending people to search for specific items, or creating a playable anti-hero. Wait, I see a resemblance here. That's what's being suggested in this very thread. Maybe you assume Nintendo would do things better. However, I KNOW that what Nintendo has already done is awesome. I don't need it re-imagined, I just want more of it. Sure you can change things a little each game, but don't force me to play in a completely different way, with a complete different game style, than I have before. That's not why I wanted a direct sequel. The "action" stages in Sonic Adventure and its sequel were great, but most of the other modes were lame, and Sega never learned this. Nintendo isn't learning it about StarFox, either. Don't encourage them not to.
As for those who enjoyed all of Assualt or are complaining about low difficulty, you're nuts, IMO. Sure, the linear flight stages were decent individually, but as an entire game, it had a poor pace. The Landmaster was bastardized. The on-foot missions were a poor reference to a poorer game, Adventures, and even with the new elements and guns, they were too slow, progression was stumbly and awkward, and stealth and tactic were unlike those in previous popular StarFox titles. As far as difficulty went, if you played the game on the most difficult setting, it was side-by-side harder than StarFox 64, so there should be no complaint about the difficulty of the game, IMO.
That's where I stand. I could be wrong, and you may disagree. Feel free to rebuttal as long as you didn't like the on-foot missions of Assault.
Oh, and I didn't look at multiplayer of the games intentionally. Any game can have a great multiplayer if it creates an environment were results can occur quickly. Games can be entirely grounded on multiplayer, take Smash Bros., for instance, however, in StarFox, multiplayer should be a secondary thought, unless it is story driven. I'd like the single player to be solidified, then take that and alter it for several people, don't go the other way around. Assault had a good multiplayer, but I bought it for the singleplayer. If they wanted to make a multiplayer game, they should have put more work in marketing the game as a multiplayer game, like Halo is marketed or something.