My ramble, the chicken or the egg, which lowers scores, the long game or the short game. While I don't think anyone would question that for Tour level players, short game is where it's at, at every tournament you hear the guy that finished second say, "If I only would have make a couple of more putts", you never hear them say, "If only I hadn't cold topped the ball on 4 and fatted it on 6" which sounds more like what the rest of us would say. I don't think there is one perfect process which you should follow to improve, but more a question of what you want to first improve and what suits your personality.
For me, I started with the short game, that was the part of the game that came easiest for me. I spent a lot of time beating balls, but spent more time with wedges, chipping, pitching, putting, and that was the first part of my game that came around. For me, I found more enjoyment out of scrambling my way around the course, hit a drive in the woods, punch into the fairway, hit it by the green, chip to within a few feet and make a putt for a bogey. The guys I played with would always shake their head and I know they were thinking, how did he do that.
For me, that was a better situation then hitting a ball in the fairway, missing a green, stubbing a chip, chip on, miss a putt and make a double like some of my buddies. I know they got frustrated because they knew they hit the ball better than I did, but never scored as well. And now with my driver woes worked out, I score better than ever, so I'm happy with the way I progressed in the game.
All of that said, I think it's really up to you as to how you want to progress, what suits your personality?