Errol, all true, but it still makes it difficult for me to suspend disbelief fully wjen I watch it. Or at least it did.
I've explained it all now.
Discovery is the best research ship the Federation has, and in order to progress their cutting edge research they realised that they needed to recruit from a diverse pool in order to find the true best in their fields. That's why we see the diverse working environment we do.
Kirk's Enterprise ten-odd years later, on the other hand, is actually one of the most basic ships in the fleet, set up with equipment that can be more easily repaired in the field without being docked at a space station. The more sophisticated ships are needed closer to Earth, in the charge of captains that aren't Kirk.
Kirk has gone down in Starfleet history due to his feats on the exploratory mission, but it starts off because they want to get rid of him. Back home at Starfleet, he is regarded as an embarrassing anachronism. He's chauvinistic, struggles with alien cultures, refuses to tolerate humans with cybernetic modifications, is appalling in hand-to-hand combat and he's always one step away from being disciplined by Starfleet H.R. for sexual harassment.
As well as the dubious achievement of hacking one of his exams, he's the most frequent flyer at the GUM Clinic run by Starfleet Medical. After a particularly embarrassing incident between him and a beautiful ambassador, Starfleet decides to promote him sideways and just send him away somewhere, anywhere. They finalise it as a mission of exploration to parts undiscovered.
And then he makes a success of it. Back home, they are very, very unhappy.
His ship is staffed with all the misfits they couldn't put anywhere else, including Spock. Once Spock had got through Starfleet Academy, they had to assign him somewhere, or risk a diplomatic incident with Vulcan. So they dumped him with Kirk and bigged up the prestige of the mission. Spock knows all this, of course, but he's studied human history of social progress and has decided it's more logical to accept the assignment than to resign from Starfleet. Kirk doesn't know this, because he's not that political a thinker. Kirk thinks the mission to boldly go is a genuine one and he's lucky to get it.