Why are moments not included in free body diagrams?

[per persson]

As I wrote, that's not quite true. Ideally there is none, but in the real world there's imprecision and flexing.
As you load the beam it flexes slightly, leading to a torque at each wall.

I remember yes.

Please, see:
https://pressbooks.library.upei.ca/statics/chapter/reaction-forces/

https://en.wikipedia.org/wiki/Cantilever

I think I understand moment better know. Consider a beam that is fixed to the wall. The gravitational force on the beam causes a moment and where it is fixed to the wall prevents the rotation of the beam thus in a free body diagram we draw a moment there. But if beam is fixed at both ends the gravitational force does not cause a moment. It is the reactionary forces at the walls that prevents rotation. Am I on the right track?

 

[haruspex]

Consider a beam that is fixed to the wall. The gravitational force on the beam causes a moment and where it is fixed to the wall prevents the rotation of the beam thus in a free body diagram we draw a moment there. But if beam is fixed at both ends the gravitational force does not cause a moment. It is the reactionary forces at the walls that prevents rotation. Am I on the right track?

There still could be, and in general will be, both a net force and a torque at each end, but the situation is ‘statically indeterminate', i.e. there is a continuum of possible combinations.
If we start the beam simply laid across two supports, then apply whatever loads, there will be just a force at each end, no torques. If we now encase the ends rigidly, there's no reason for any torque to arise.
Starting again, encase the ends before applying the loads. Now, flexion in the beam will lead to a torque at each end.