+1 to the point that both schemes result in very different gameplay.
In my experience, a Gamepad Scheme (Movement Keys + Action Keys) leads to more precise platforming, as precise control of the character's position is the main skill at use.
Free Aim (Mouse + Movement Keys) shifts focus to the aiming, and is better for shooting-heavy games where aim is important, but with light platforming.
Of the games I've tried with both schemes, I've always had trouble with those that either tried to do both things, or had the wrong focus.
As a simple example, I have an easier time platforming in Trine using the Knight (whose mouse controls only matter in close combat, so I forget about the mouse when jumping around) than with the Rogue, who is ironically the platforming-focused character. The grappling hook helps though, and I feel it was a nice addition to alleviate the control issues.
Again, think hard what works best for your game, and go for it.
As for making the control scheme configurable... I'd only do it if the game had the same amount of heavy aim-dependent shooting as tight platforming, so both schemes are at an equal advantage/disadvantage.
Another alternative would be to change the control scheme with different weapons. Say, if the player holsters their gun, they go into platforming mode, where the character can hold on to ledges, Prince of Persia style, and the mouse crosshair is just gone.
I wonder how free aim would work on consoles or tablets?
I seem to recall that there were old games using free aim in consoles. Basically, the up/down d-pad input controlled the firing angle.
Nowadays, the thumbsticks would make directional aiming easier, I think.
As for tablets, you can tap anywhere you want the character to shoot at. The only problem is making sure the player's hand does not obscure the screen. Maybe having the player tap at the screen edges?
It's tricky, but I guess it's doable.