Could the acceleration of a projectile be represented in terms of a radial and a tangential component at each point of the motion? If so, is there any advantage to this representation?

Yes ,the projectile motion can be described in terms of tangential and radial acceleration. For a projectile motion the tangential acceleration would be zero accounting for the fact that there is no change in horizontal component of velocity while the radial acceleration would be equal to g .
This method is advantageous only when the center of the circle corresponding to the arc traced by the projectile lies on the surface of Earth. This will allow one to use radius of the circle as the maximum height of projectile and diameter of the circle as the range.