If Fairy Tail was consistently well written, Natsu and Gajeel would've developed a friendship (outright friendship, not the "deep down they're friends even though they literally never act like it" shit) and become a frequent team, it's a no-brainer. All the set up for it is already right there in the story, it just fails to follow through. Their rivalry's redundant, for one thing, since Gray's already well established as Natsu's rival, and for another, after Gajeel joins the guild, there's no clear reason for their fights. It starts off like lingering hostility from the Phantom Lord debacle, which is just more reason they should've made up during Gajeel's character arc!
That was the trajectory for so long, too. Immediately after the Phantom fight, Natsu offers a truce by asking about Gajeel's magic and the dragons; he straight up says he wanted to be nice now that their fight was settled. He doesn't hold grudges, that's one of his core traits. The literal title of that chapter is "Like-Minded", they're intended to have shared qualities from the get go.
It makes sense Gajeel shoots him down bc he's still incessantly antagonistic at that point. But after he joins Fairy Tail, they repeatedly have to team up, and Gajeel is consistently the first one to accept it and insist. Against Laxus, he's begrudging about it, but he puts up way less resistance than Natsu. And it's subtle, but Gajeel's also the one actively pursuing chances to combine their attacks and calling those shots. AND, despite any competitive, mid-battle bickering schtick, he's willing to give ground to make an opportunity for Natsu to win. He doesn't miss a beat either, it all comes naturally right off the bat.
The Edolas fight with Wendy continues on this track where both of them make less of a fuss about cooperating this time- Natsu only makes one short gripe and Gajeel's down to none, he's 100% on board with working together. And again he's initiating joint attacks and giving Natsu support so he's able to take the offensive. He had already thought of the plan to combine their roars! Gajeel keeps demonstrating that he wants to cooperate with Natsu when they team up, like how Natsu originally tried to get along!
The GMG double battle should have been the turning point. The lead up to it's already there, and even in the first part of that fight, they keep using those methods and make a good team. It's very small, but I think Gajeel offering to split the difference when they bicker shows that he, for one, is willing to compromise, and he's not really invested in butting heads Natsu. It was the perfect chance for them to officially bury the hatchet and show off their combined power- then Natsu shoves him down a chasm : )
It could've easy done something else, like maybe Natsu wants Gajeel out of the way so he can fight Sting and Rogue alone, but Gajeel calls him out for putting petty insults over doing right by their family, because they're here to show the strength of their own guild and make them proud, not to beat their opponents in the most humiliating way possible. It'd showcase how Gajeel's changed and embraced the guild's camaraderie, and for Natsu to recognize that he's not fighting for his family when he won't fight alongside his family would spark some development he never got. It could echo their debate over teaming up against Laxus, coming full circle. Both of them agreeing to work together and beating Sting and Rogue as a team would suit the idea of the bonds of friendship overpowering personal ambition and give some actual substance and sincerity to it. Just anything other than him kicking Gajeel to the curb.
I find it so forced and unnatural that Natsu just discards Gajeel's efforts to cooperate after everything else. Like why? It's just presented as a way of getting back at Sabertooth, which makes no sense for a character who's always puts his guild first when it matters. And after Gajeel's tried so hard to make amends and earn acceptance, the writing is just so sloppy and it's stupid as hell. Bc it could've added much needed depth to Gajeel's arc too! When he's initially so prickly, it's very intriguing that he also readily joins up with Natsu. It'd be insightful to reveal Gajeel was taught to fight on his own, but once he finds his guild he enjoys and is good at fighting with a team. It'd be fun to see him find the means to wield that! Like why write all those scenes of them learning to set aside hard feelings and test their potential power as a team, then not deliver? If it did, then their fight in Tartaros could've been really cool showing the fruits of their progress instead of throwing it all away, but instead it's just dumb arrrrargrrghhh!!! GIVE US THE NATSU AND GAJEEL FRIENDSHIP WE DESERVED