Grunt's problem was that something he was seeing as his problem, actually wasn't a problem.
So there was nothing to easily 'fix'. ...Except Grunt's perception of his state.
Through embracing the fight for survival, he taps into the core biological motivation for every Krogan. On the Normandy he is extra safe and he doesn't like that. It breeds 'weakness'. Every second he is not training against a real enemy, is a second that he may not be prepared for stronger opposition.
The krogan must always have *some* enemy to fight, even to a minor degree, unless we want to genetically change them. And Okeer certainly did not change this trait for Grunt, but if anything emphasized it.
While acceptance from other krogan is nice and all (and Grunt embraces it in ME3), what Grunt is looking for is the abstract permission to do the violence he does. With the Rite of Passage, he gets that.
The lesson of the mission was that 'sometimes you gotta roar'. Instincts when honed and trained can sometimes empower you to do exactly what must be done, even if you don't intellectually understand it. In fact, understanding it emotionally will give a far more visceral understanding than anything academic. That's what Grunt experiences. He is now truly Krogan. For better or worse.
This carries on in ME3, where Grunt takes his fight for survival to heart and never, ever, ever gives up against the horde of rachni. And he survives to fight another day. Sadly, off-screen.
Shepard can also take this fight for survival to heart in his future decision making.*
*For the record, I think pretty much every henchman carries lessons in their stories, and not all of them outright focus on something like, say, Breath Destroy. For example, I'd consider Samara to give Control lessons, and Thane to give Synthesis lessons, even while both might still advice you to Destroy. Grunt is just a more basic character, to more directly carry the ideas of war, strength, competition, survival, etc., stuff that Shepard's own main story is also about.