On a principal level, I don't think that jack-of-all-trades type of characters are necessarily a good thing in the long run if there is no major sacrifice to them. DnD and PoTM both strive for role-based party mechanics, where each character has a specific role to fill. You've a front-line tank, a buffer cleric, a trapmaster ranger, a lock-picking rogue. By building this kind of a system, where a variety of characters come required, we encourage diversity in builds and partying with other players.
To me, personally, it's much more desirable to limit than to empower the variety of capabilities of a single character. I know this is way beyond your suggestion, Dreamless, but when classless approach is taken to the extreme, eventually party roles are fully gone. This is what I feel is wrong in otherwise awesome games like Path of Exile; Partying feels more like a bunch of characters striving for maximum DPS who just happen to be in the same map killing same monsters, rather than a bunch of characters filling their own roles and synergizing well together.
DnD, in its adventure mechanics, is all about synergizing with the party.
The in-character explanation for the skills your class offers ties to what kind of activies have been leading to your level. A fighter, who has full BAB and more hitpoints, is assumed to have been mostly practicing fighting. Hence, they don't have many non-combat skills available. Rogue on the other hand, who does not have full BAB and as many hitpoints, is assumed to have been doing those more rogueish things. Hence, he has more rogue skills available. Here game balance also comes to question.
All that said - As it is, multiclassing is pretty powerful. Particularly in the case of fighter, it feels more difficult to come up with a reason to not branch to rogue (or another full BAB class than fighter) than to branch. Many jack-of-all-trades, master of none builds actually wind up to jack-of-all-trades and damn good in half of them.