What is violated here

I am at the moment trying to do something different in the classical mathematical derivations of the equations for rotations and trying to find out why this would not yield a correct result.
This time I do this:

Consider an arbitrarily shaped rigid body rotating about an axis. Let us find out a relationship between its angular acceleration and the external forces acting on it.

For a small masselement of mass m_i we note that the tangential force on it is given by:

F_tan-i = m_ir_iα
To derive the torque equation you normally multiply by r_i and sum up all the torques to get:
Ʃτ = Iα
You can trust that Ʃτ is the sum of external torques only since a simple argument shows that all internal torques cancel.

But let us instead multiply by r_i² and sum up to get:

ƩF_tan-ir_i² = (Ʃm_ir_i³)α

Then you can still use the same argument for the internal quantities defined by the right hand side cancelling out. Yet, it does not yield the same result as the correct equation Ʃτ = Iα.
What is violated by this approach?