I disagree about the "complete" do-over part. There are little pieces here and there wrong with my swing, and one at a time I'll pick them off and the swing will improve each time. I think that is true about everybody.
And as far as most amateurs not having enough time to get better ... well, then there is really no good answer. If you don't have time to practice the things you need to work on, you will not get better. If somebody really wants to get better they will make time. (I don't have a lot of free time right now with little toddlers running rampant at home all the time, so I practice at lunchtime during work 2 or 3 times a week.)
And if I was you, since your short game is already your strength, I'd start working on something else. That's why I practice my long game when I practice. Specifically because it is my biggest, most glaring, weakness. It's not hard to get the ball on the green from 50-60 yards and in (or whatever you would define as the "short game"), and putting ain't that hard either so why waste too much time on them? I mean, I don't practice putting at all and over the course of those same 11 rounds I've averaged 31.5 putts per round. How many strokes could I possibly save by practicing putting and chipping? One, maybe 2 a round? Conversely, how many strokes can I save by figuring out how to keep the ball in the fairway? Like I said, 4 penalties a round, figure most are lateral hazards, so I'd estimate somewhere in the vicinity of 6 shots a round, just on those alone. Nevermind the fact that the better swing is going to also bring more of the less wayward tee shots into the fairway, its going to bring more of those approach shots onto the green for those precious GIR's. Once I start nailing fairways and greens, then it's time to really fine tune that short game because it'll be those last few strokes needed to get down to scratch.
Thinking of practicing short game, short game, short game, to lower scores reminds me of that silly Prilosec OTC commercial where Larry the Cable Guy says "taking medicine for heartburn after you have it is like checking on your burgers after they're burnt." Same is true for short game practice when your struggling to get to the green. So what if you are good at getting up and down if its always for bogeys and doubles?
That's what I thought last season and my short game went straight to hell. I learned a hard lesson. There is no part of the game that you can always have an innate grasp of and I prefer to KEEP the short game as a strength. And I totally disagree with your thoughts about practicing putting. Now, there is a place you can really shave strokes.
I always hit drivers and irons when I practice, but I spend more time on the short game. I hit drivers on the range before I play and try to keep a single swing thought to take to the course.