I understand what you mean, however, I think what most other class players do is have fun within given parameters, and can readily start a new character without any sense of losing out on anything. We Jack of Traders want it all! We want the ability, within one playthrough, to say," I'm tired of being a Mage, let's try a Stealthy game play, or even a Combat role"....at the flip of a dime!!
Hmm. For me, jack-of-many-trades is more a case of choosing playstyles (or "classes", though I rarely think of them that way) with a wider array of skills, rather than one focused on, say, 4 skills. And yeah - feeling free to dabble in other skills when needed or when it seems appropriate. But when I want to switch to an entirely different playstyle (like a real
mage, rather than a melee, or even a melee with some spells), I make a new character.

Spoiler My first Skyrim character, for instance, could be labeled a "Ranger".... primary combat was sword & board, light armor. But also used archery - unperked - to open fights and hunt animals, sneak for scouting, lockpicking because every adventurer needs that for dungeondiving, smithing and a bit of alchemy to maintain one's own equipment, and speech because it naturally rose from dealing with people. And Restoration for healing. Ten skills spread among all three archetypes, but still a basically defined class/playstyle.
(I'm more into the full phrase.... jack of all trades,
master of none.

)
Of course, this was reinforced in Oblivion, due to the whole "efficient leveling" thing, where one played the game off minor skills, and only used the major/class skills when you needed to force a level gain. So I'm really used to just ignoring classes and major skills.