A few things... We know there are at least ten "fleets", which term seems to be more a matter of "ships operating under the operational authority of such-and-so Admiral out of Starbase yadda-yadda for XYZ chunk of Federation space" than "bunch of ships all going to this place and doing that thing at once". Except in time of war when the nominal fleets muster to converge on the hot spot. I use "Court Marital" as a good example. The ships on the chart in Stone's office were all operating out of Starbase 11, but on their own individual missions rather than in a single massed flotilla. A good analogy would be the US Pacific Fleet and Atlantic Fleet. There are individual ships and battle groups and task forces within each, and it'd take something major to mass the whole fleet in one place.
We know that, at least as of the Dominion War, there were a minimum of ten fleets in Starfleet. The Second Fleet was the one that covered the region including the Bajor Sector, and many Starfleet assets were retasked to it during the late buildup and early stages of the actual war. Elements of the Fifth and Ninth Fleets were added to it for the retaking of Deep Space 9, with the station becoming the new HQ for the Ninth Fleet for the remainder of the war. The Fifth was retasked from defending Bolian space, and the Sixth Fleet redeployed to compensate. The Third Fleet went to bolster the defense of Sector 001 after the Breen attack (probably reinforcing the First Fleet), the Seventh Fleet was the primary roadblock to core systems like Betazed and Vulcan, and got repeatedly mauled by the Cardassian-Dominion forces. The Tenth Fleet was behind them, defending Betazed. The Fourth and Eighth Fleets aren't mentioned and were likely on the far side of Federation space, being deployed to cover the coreward parts of Alpha and Beta Quadrants (Ferengi space, the First Federation, the Romulans...). We consistently see majority portions of those fleets in the couple-hundred-ship range -- many of them ships we normally see serving as behind-the-lines science or transport vessels. And the combined portion of the Second and Fifth Fleets going to retake DS9 (the Ninth was a day behind) numbered roughly 600.
So it's easy for me to extrapolate that out to about 5,000 or so active vessels in Starfleet circa 2375, including non-combatants and those being repaired. Out of those, there are nine Galaxy class starships, and only the best of the best get there. This also doesn't include non-Starfleet "civilian" ships flagged to one or another member system. I imagine there are all levels of aspiration and ambition amongst 24th-century folks (human and non), and many are content hauling cargo, many are content getting whatever billet will let them study nebulae or pulsars or what have you. Comparatively few might even be interested in trying to get onto a deep-space Explorer that can, should the mission dictate, head out away from known space for a good fifteen years at a stretch.
Picard did something between the loss of the Stargazer and when we first meet him in "Encounter at Farpoint" to give him a lot of clout, enough to get the new Enterprise and handpick his command crew. They offered him a promotion to the Admiralty and the position of Commandant of Starfleet Academy that same year. Riker had been offered his first command when he was still a Lieutenant Commander (probably the rank Picard was when he had to take the bridge of the Stargazer, by the way), and he'd only been a full Commander and on the Enterprise for about a year and a half when they made their second offer. It was only when they made their third offer another year and a half after that that he was in someone's way on the Enterprise for the first time. And Shelby's ambition was out of the ordinary, as Hanson commented on it and both Riker and Picard were rocked back by it. I get the feeling for those top-of-the-line ships, you have to really want it to get there.
As for the other command crew oddities... Given how many Chief Engineers they went through in the first season, I wouldn't be surprised if Picard had tapped Geordie for the position, but he had to finish out one more year as Lt., j.g., before he could be given the position. It was also odd that Yar was both Security Chief (Operations Division) and Tactical Officer (Command Division). Worf was basically Deputy Tactical Officer, and took over as both when she died (even though he didn't get promoted to full Lieutenant until season three). Not sure why the Operations color would take precedence in that sort of doubled position. Similarly, why was Spock in Sciences blue when he was also the First Officer (Command Division)?
Anyway, data to ponder...
--Jonah