Why is the electric charge on a solid object always explained in terms of an excess or deflcit of electrons?

There are many reasons.
The first is that we are super lucky and the positive charges of the atoms (the protons) has exactly the same charge as the electrons but with the opposite sign.
So to say that an object has a missing electron or an additional proton, from the point of view of the charge is the same.

Second, what is moving in the materials are the electrons. The protons are strongly bounded in the nucleus and to remove or add them is a complicate process that does not happen easily. While to add or remove electrons can be sufficient to pass your object (for example if it is plastic) on wool.

Third, if you change the number of electrons you ionise the object, but many fundamental (especially chemical) properties are unchanged.
If you change the number of protons, you are moving on the periodic table and changing the element, and this does not happen frequently in the electric phenomenon (because they involves electrons in general).

So the short answer is that everything is described in terms of electrons because usually those are the particles doing all the electric business.