Why do we neglect torque caused by tension of curved part of rope in massive pulleys?

Consider an infinitesimally small section of the rope, at an angle $theta$ (measured clockwise) from vertical upward direction (see picture at the bottom)

Let the tension be T, on one side of this section and (T+dT) on the other side, both acting in opposite directions (exerting opposing torques, counter clockwise and clockwise respectively).

Net torque due to this infinitesimal section of rope is $$(T+dT)R - TR = (dT)R$$

Integrating this between end points to get the over all net torque $$(T_2-T_1)R$$

The professor is doing something tricky, which they should have mentioned explicitly.

In order to find a total torque, we need to specify what system we're considering. Suppose our system is the pulley. In that case, the external forces are the force from the support, gravity, the normal force from the curved part of the rope, and the friction from the curved part of the rope. The first three forces provide no torque because they go through the center of the pulley. Only the friction provides torque.

But the professor didn't mention friction at all, and they did mention tension, which is not even acting on the pulley! What's going on?

The trick is that they're not considering the pulley. They're instead considering the net torque on the pulley plus the curved part of the rope. The point of this choice is that now all the complicated forces we don't want to bother with (the tensions of different pieces of the curved part on each other, the friction and normal forces between the pulley and curved part) are internal, so they don't provide a net torque. Now the only external forces are the force from the support, gravity, and the tensions from the two straight segments of rope. Only the latter two can give torques, so those are the forces the professor considered.

But why is it allowed to change the system? Because the point of taking torques here is to set $tau = I alpha$ to find an angular acceleration. And since the rope is massless, the value of $I$ is the same whether you consider the pulley as a system, or the pulley plus curved rope as a system. So you can use the latter to compute $tau$, and thus compute $alpha$.