I, of course, have also played 'make a ball - keep shooting' ever since I learned the game 25 years or so ago.
Personally, here is why I have grown to like the re-rack way better:
When playing someone and you see the bottom ball leak out and go over right in front of your opponent's hole, leaving them an easy first shot at their hole with potential to run more, you wonder, did I hit that bad? Did I hit the wrong ball? Did they rig the rack on me??
Obviously the breaker has the chance to check over the rack and make sure it's good. A lot of times we don't do that (myself admittedly) because you don't spend the extra time or maybe you don't want to suggest you don't trust your opponent, or maybe you do look and don't catch whatever they may have done to it.
The point is, if I get to rack my own, I will never have to worry about that. i am 100% confident that I am not going to rack it bad for myself, at least not intentionally lol. So now it puts all of it on me. If I get to rack and break, and it comes out bad, then i need to work on that part of my game.
That being said, and since people DO work on their rack and break, I am sure that there are ways to manipulate the rack so that you make a ball or make it turn out a lot better than if your opponent were to rack. I don't do it, but some guarantee do.
So if you play re-rack when a ball goes in or break and done, you don't ever have to worry about it. IMO it takes any unfairness out of the equation. Also, this isn't 9-ball, isn't the break for our chess like game meant to position balls to our side while keeping the CB relatively safe?
I am all for shot clocks or anything else that may speed the game play up, but I don't think continuing to shoot if you make one on the break is part of a good solution for that.