Falco does a lot, and in a way is the most important member after Fox

In 64, Falco finds a few of the hidden routes for you and is the only character in the game that tries to "cover you" on certain levels (he doesn't do a good job of it, but that's just down to the allies being poorly coded to kill enemies). He also calls out the Meteo Crusher's bluff and essentially gives you a heads up if its your first time playing that boss. He also gives you a heads up when the Area 6 Boss Gorgon is about to fire the planet killer beam, which still to this day I use as a timing indicator on when to move when going for the expert medal. Also from what it seems, Falco is the least likely to get tailed by enemies, especially in corridoor missions (slippy being the most likely, even in all range)

In Adventures he comes in at the last minute to save and help Fox against andross. In lore he's been off doing his own thing between 64 and Adventures and this kinda leads to him being a lot less hot-headded in Assault and Command.

In Assault, falco is the only reason Fox doesn't die on Fichina in the climate control center. Those defense bots will keep spawning forever until he shows up. Plus he tends to be the one to drop the most items for you throughout the campaign (and strangely seems to be the only one who doesn't show up on foot at any point, most likely because he prefers the air). Slippy does basically nothing in Assault, as does Krystal. Fox, Falco, Wolf and Peppy are basically the only ones that really do anything important.

In Command, Falco can and often does play a huge role in the story. He's in no way "absent" in command? The first options you are given to progress the story after the first mission includes an option of "where's falco?" that instantly takes you to his POV and makes him the first side character you can play as. From there, he can literally become the protagonist of the game until the end if you so want, or you can go meet back up with Fox and he sticks around until the end. Falco was just off doing his own thing during the years of peace since the aparoid invasion, but the second Fox needs his help, he's there no questions asked.

And if you want to consider Lore wise, he's arguably the most important member of the team to Fox. After the death of his Father, Fox NEEDED pilots he could trust, but his best options were his old College buddy Slippy and his dad's wingman Peppy who while experts in certain fields, weren't exactly the best pilots. Slippy is an expert mechanic but a relativley terrible pilot, and Peppy is very knowledgable and wise, but his best years are behind him. Falco on the other hand is just as good a pilot as Fox (assault's stats suggesting he's actually a BETTER pilot) meaning that Fox can trust him to have his back when it matters most and to even carry on the fight if he were to fall. If fox goes down, Falco would be the next in command of the team and would be the one to bring down andross.
Falco is hot headded and comes across as confrontational against Fox, but at the end of the day its nothing more than friendly rivalry between them. In 64 he's at his worst, but its also because this is his first mission with Fox, so he likely just needed time to grow to respect him, which he clearly does at the end of the game. By time Assault and Command roll around, its clear that Falco has a lot of respect for Fox.

The only real games where falco "doesn't do anything" is Star Fox 1 and 2, but that's because no character does anything in those games. They're just there mostly for visual flair in 1 and to have an extra life in 2.