Braced right leg at address, leisurely transition, downswing that starts with a shift onto your left foot combined with a hip twist followed by a relaxed swing. Do this with the ball setup aligned your left armpit (or inside your left heel) - this creates room for your weight shift.
Until you learn to begin your downswing swing with your lower body you will struggle because leading with your lower body creates an inside out swing path - not anything you do with your arms.
It is true that it is possible to create an inside out swing path by controlling the angles of your left arm and wrist (while maintaining a relaxed right arm). This will give you a good short game. But, you will endlessly struggle with your long game until you get your lower body into the mix. So print the first 2 paragraphs of this post and take it to the range year after year until you get it right because this is the answer you are searching for.
Looking back I'd say the greens at Oakmont made for a less entertaining tournament. Golf is most compelling when the players take on, rather than avoid, the challenges before them. Super-fast greens turn a golf tournament into a driving contest...with a crap shoot on the side.
I think a better correlation would be the better club head position at impact, the better the score.
You see a lot of poor (ugly?) swings out there, that are still carding decent scores. The swing supplies the power,but the club head position at impact sends the ball where it's going.
I have held my head still for 30 years of golf. I know have two known and good pros who allow me to move the head. One directly tells me to not restrict head the other tells me to bump the hips and ignores may head sway to the front. I gain 2 clubs on launch monitor.
Keeping the head still makes me turn in a barrel, keeps me from bumping left and costs me yardage. Right knee goes out right elbow stuck behind, right heel lifts early, legs don't cross in back view.
Holding the head in one position forcefully can very much restrict your swing.
If by "better swing" you mean a swing that is more mechanically sound and hits on all the major points required to consistently strike the ball with accuracy, then yes there is a strong correlation. GIR and nGIR are the biggest stats when it comes to lowering your scores, and a technically better swing will produce more nGIR and GIR.
If by "better swing" you really mean "better looking swing", then the answer is no. Arnold Palmer, Jim Furyk, Jim Thorpe, etc. have all made it work quite well with a swing that most would consider ugly.