Yep, you can't build up one without diminishing the other.
If you want to build the two up to be such close friends, like "brothers", then there's no way that they could maintain such a huge rivalry because, as you said, such a huge, even obsessive rivalry will only lead to conflict.
If you want to build the two up to be such a huge, even obsessive rivals to the point of conflict, then there's no way that they could also have such a close "bond or brotherhood" because having such a bond would mean that they actually DON'T want to be in conflict.
A bunch of other such rivalries have, as usual with a bunch of things "Naruto" tries to do, done it SO much better. Like, look at Yugi/Atem and Kaiba in Yu-Gi-Oh!. Throughout the WHOLE Duel Monsters series, you have Yugi/Atem trying to convince Kaiba to be their friend and work together and such (though, at times, it sounds more like them trying to force their beliefs onto Kaiba...much like Naruto with antagonists), but Kaiba always stuck to his guns in being adamant about just wanting to defeat Yugi/Atem in a Duel. The furthest they ever go is simply having Kaiba more openly respect Yugi/Atem as fellow Duelists, but never do they actually become "friends".
Agreed. If anything, I kinda wish that Naruto and Sasuke's rivalry and bonds had been more akin to Goku and Vegeta, which is really one of the best examples you can think of with that.
Given their connection begins very antagonistic due to Vegeta's actions during the Saiyan Saga, as well as his direct AND indirect hand in killing many of Goku's friends. Plus we get the fact Vegeta assuming he's immediately stronger than Goku due to his being the prince of Saiyans while Goku was just a common foot soldier, only to be surprised, and also irritated, wanting to find means to why he was able to deal blows and such like he was by Goku while on Earth.
And we see that continue in the Saiyan Saga with Vegeta doing all he could to try to become stronger, only to once again be passed by Goku and his growth when he handles Recoome, someone who nearly killed Vegeta, Gohan, and Krillin, like it was nothing, leading to Vegeta's envy growing because of Goku's growing closer to the status of Super Saiyan.
It then escalates more and more near the climax of the Namek/Frieza Sagas when Vegeta is killed by Frieza and begs Goku to avenge their people, as well as his wanting to remain on Earth to allow himself to figure out how Goku became a Super Saiyan, and even when he does, it still isn't completely enough in his mind, in spite of getting what he wanted to be able to catch up, and in his head, to surpass Goku's power.
Then we see it die down slowly when Goku sacrifices himself to stop Cell from destroying Earth, leading to a calmer Vegeta in the beginning of the Buu Saga, one that has been training his and Bulma's son Trunks as well as continuing his own path of training, until he witnesses how much stronger Goku has gotten from training in Other World the seven years he'd been dead, and being able to defeat Yakon like he did, which led him to make his Faustian bargain with Babidi to gain the Majin power, leading to his and Goku's penultimate battle with his admitting why he did it even as proud as he was to never ask for help.
With this also culminating when Vegeta sacrifices himself to try to kill Buu, even giving Trunks a hug beforehand, then his return and his anger at Goku for hiding the fact he could go Super Saiyan 3 the whole time during their fight, leading to the final battle with Kid Buu, where Vegeta comes to grips with the fact Goku is simply better than him.
Then with Battle of Gods and leading into Super, we see even as Vegeta has acknowledge Goku being better than him, he's still motivated to surpass him, even as Goku manages to get used to his Super Saiyan God form, with Vegeta following behind and his gaining it as well as both becoming Super Saiyan Blue, but we also see how Vegeta's ego still is a fatal problem in Resurrection "F" when he doesn't take his chance to finish Frieza off because he wanted to make him suffer for all Frieza did to him, allowing Frieza to destroy Earth in a pyrrhic victory.
But overall, that's what makes Goku and Vegeta's rivalry and connection strong is that even with how it is, they can still see each other as friends while retaining that healthy rivalry that allows them to both become stronger in their own ways. I feel if that was done for Naruto and Sasuke, it could have been so much better than what we did get, considering we did get some moments of them actually acting like friends, but it was rare and inbetween, leading it to be unbalanced and how we got it to be more of an obsessive rivalry kind of thing.