In all books and texts I've seen so far, Newton's Second Law is used to prove that the net torque acting upon a system of particles with respect to its center of mass is equal to the rate of change of this system's angular momentum with respect to its center of mass. At this point, a crucial assumption is introduced: the internal force pairs obey the strong form of Newton's Third Law. It is then shown that the rate of change of the system's angular momentum is in fact equal to the external net torque acting on it. This result is then applied to rigid bodies without any caveats, and, surprisingly?, it works.
The issue with this is: for a rigid body to remain rigid, Newton's Third Law cannot hold in its strong form. A simple example shows this: picture a rigid body system made up of three particles, initially at rest and positioned along a vertical line, with the middle particle separated by some distance $d$ from the other two particles. A horizontal force directed to the right is applied to the upper-most particle, and another horizontal force of the same magnitude directed to the left is applied to the middle one (it's as if the middle one was a pivot). The linear momentum and angular momentum theorems tell us that both the upper and bottom particles will experience acceleration in the horizontal direction. This must be the case for the rigid-body condition to hold. So the bottom particle experiences net force. Since there is no external force acting on it, that force must originate from within the system, i.e., the bottom particle is acted upon by either the top particle, the middle particle, or both particles. Since the line of action from the bottom particle to those other particles is vertical and the net force it experiences lies in the horizontal direction, Newton's Third Law does not hold in its strong form.
All this means that internal force pairs in a rigid body produce torque. So this must mean that it is not true that the net torque acting upon a system of particles (both torque from the surroundings and from within the system) with respect to its center of mass is equal to the rate of change of this system's angular momentum with respect to its center of mass, if the system is a rigid body. But Newton's Second Law implies such. So Newton's Second Law, in turn, is also violated for rigid bodies.
The real, physical, world, deals with this easily: rigid bodies don't in fact exist. Any body undergoes stresses and strains when under the action of non-body forces.
What is bewildering to me is how the mathematical model of a rigid body seemingly doesn't make any sense in the context of Newtonian mechanics, and yet it still produces precisely the results we would expect. It's as if we start with a set of laws, deduce some properties of systems after applying some hypotheses, then naively translate this to rigid bodies, which, apparently, should work under a different set of laws, and yet it all works out in the end mathematically. How does this not break apart? What is going on? Have I made a gross mistake somewhere?