I haven't been following this thread too closely. However, if the question is "why take the Lagrangian to be quadratic in the velocities", the answer is that we do so in order to get equations of motion which are linear in the velocities, no more, no less. If you are brave enough to drop this requirement, then more general Lagrangians are on the table that satisfy whatever symmetry requirements you choose to impose, as others in this thread have noted.