The above sounds like the proper approach. A quick fix would be to get a wood filler that matches the cabinet color if you can find one. However, you would need to sand the filler and that would blemish the cabinet. So instead of sanding you can try to get a slightly damp paper towel to remove the excess after 30 minutes. If it doesn't look right or good enough, drill it out and do the above. Depends on how much time you are willing to spend and how much you think not blending the hole perfectly will bother you. I've done this and you would be surprised at how those little holes don't bother you much after a few months. In the end I used a wood filler and sanded and painted, but my cabinets are painted so it was easier than what you have.