In older games, where you rolled statistic dice in order (and I'd have to include BRP games like Call of Cthulhu, the chart-filled Rolemaster, and all the early D&D games here), and made a character suitable to what you were given, there wasn't much use in rebuilding things from scratch. And the "game" nature of the game would make something like that seem almost like cheating, anyway.
Thus, in the old days if your only high Stat was Strength, you were probably headed for a Fighter.
(I'm sure out there there is at least one proud player who is fondly regarding their framed thirty year old character sheet with an Int 11 Str 18 Mage).
However in newer games you often build the character you want, either by re-arranging rolls or by purchasing your statistics with a pool of points. One of the strengths I see in these systems is the ability to alter the mechanics of the character while keeping the same character.
In R's excellent Margreve Forest Pathfinder game my character, Sasha, was built as a Verdant Sorcerer. But it became clear that I was playing a Druid. So, somewhere around lvl 3 we "re-skinned" Sasha, and he was a Druid for ever more. This is a good thing, my character met the concept better, blended into the game better, given the chance to do the same all over again I still would have changed to Druid.
But what about that time he cast that arcane spell? Doesn't matter. It's magic, who knows how it works?
In my current Kingmaker game one of the players who is playing an Alchemist may re-skin to a Sorcerer. I'm all for it. Not for the reasons the rest of the players may be -- their characters have been on the end of Alchemical splash damage a time or two too many, and Alchemists are phenomenally "selfish" casters, most of their spells having persona effects only. But neither problem really concerns me.
Rather, I want the player to be happy with what they're playing. If the player would prefer a Paladin or a Dwarven Rogue then the current character could retire. But if we keep the trappings and effects of alchemy, potions, and so forth I'm more than happy to allow the player a chance to re-do the character.
And if after a couple sessions it doesn't work out and he wants to go back? That's okay with me too.
It is in places like this that the divide between games of the older styles and newer styles begin to yawn wide beneath my feet.