No, it's just the positioning of the index finger that is the question. You see, if I play the old way with the thumb wrapped up the side of the neck, then I am forced to place my fingers leaning at sideways somewhat of an angle when I make various chords with them.

This causes the index finger to engage the fingerboard leaning...which is a structurally weaker position than engaging the fingerboard straight on...as I can do when my thumb is in "classical" position, because it gives my fingers more reach.

When I play in classical position, the index finger does not hurt at all, because it's not leaning over at an angle, therefore it's not stressing the first knuckle (which is where it hurts when I don't play in classical position).

A simple matter of geometry. Your fingers work better pressing   straight down on something than they do leaning to one side and pressing down on the same thing. This would also be true of a pair of plyers or any kind of gripping or pressing tool. You lose strength if you don't grip or press in a balanced way...and that's what classical position is all about...equally balancing the stresses on the hand and fingers.