Drop a stone into a pool of water and look at the ripples produced. Do you see it "accelerating" from zero velocity, or does the wave already move at a certain speed when it is produced?
This "magic" is already all around you, even in such ordinary observation as water waves. It is so familiar even if you are not aware of it.
Your question on how something can exert a force, or needs to have a force exerted on them, and yet not have mass, is why I often tell people that you can't learn physics in bits and pieces. You see, way back in the 1800's, before the concept of photons even came into existence, light was known to be a classical wave, described via Maxwell Equations. There was no concept of particles here.
Yet, even back then, they already have a knowledge of "radiation pressure", meaning they already have a description of light pushing against something, i.e. imparting a force. Yet, nowhere in the formulation of light at that time was there ANY involvement of "mass" for this entity.
Texts on classical E&M still have this concept. So already, the insistence that having a force must equate to having a mass is faulty.
Zz.