Wood – How to hide exposed countersunk screws in kitchen cabinets side panel

II would say you are fine (and you may have been fine before, who knows). You have replaced the screws that hold the cabinets to the studs, and those are the ones that hold the cabinets up. The ones in the face just keep the faces flush, so if one ever breaks, do the repair then. Hope this helps.

When installing cabinets, the "go-to" method is to "hang" them on a rail that you install first. The idea is that the rail, being much lighter and less bulky, is easier to put in place, get level, and then screw into the studs. Sometimes it's just one rail, sometimes two; depends on the cabinet system. The rail should be securely screwed into every stud along its length, to provide a solid anchor for the cabinet's weight.

Then, the cabinets will have a complimenting hanger hardware that will allow the cabinet to hang on the rail(s). That's enough to support the weight of the cabinets and what they hold, but you should still secure the cabinets to the rail, and thus to the wall, with screws through the backing and/or the back edge of the cabinet side panels. Again, how this works depends on the cabinet system; most cabinets have a pretty flimsy backing, but pre-drilled holes to put screws into the rails behind. Then, you put some trim around the back to hide any gap caused by the rail hardware, and you're done.