Well, one case is with regards to gravity. Since this force acts on all the parts of the object, it amounts to working on "all of it" so we can treat all of it as moving due to this force. This is especially so in the case of a uniform gravity (field), as is often taught initially (i.e. all free-falling bodies move with acceleration g). When it comes to gravity between spherical planets, one needs to invoke Newton's theorems that each such body generates and is affected by gravity like a point mass.
Another elementary case is collisions and motions in a straight line of balls (or, better, frictionless discs and so on). It is clear from the definition of momentum why the body will move in a straight line as a whole: each part moves on its own, so the whole body does. And collisions are momentary, so you can "ignore" the complex process of the collision itself and focus on the outcome. Whatever the total internal forces do during the collision, we're just interested in what happens after them to the motion of the whole ball and hence the motion of its center.
This really leaves only things like pulling, pushing, and friction as the more problematic still-elementary cases. A real explanation of these would require moments. However at the elementary level before reaching moments, one can say that we're ignoring the rotation of bodies at this stage. So consider for example a force pulling a block of wood horizontally. We can imagine the force pulling from the center of one end, and the block as being composed of a set of springs, so that the force stretches all springs throughout the body until the entire body is pulled. Which I think makes it intuitive that the force is pulling all the mass, and hence we can track the block by tracking it as a point mass.
I think the above should justify things enough for most cases encountered in a basic Newtonian mechanics course, before it reaches rotational energy, moments, and so on.