If you are RFF you should have your left palm up, right palm down, it will make all of your pulls stronger.
I can't speak to what those skiers are doing but I have a couple of things that work well for me
1) be in the right body position BEFORE you pull out. The first thing I do when I get pulled up is get myself set in good body position well before my pullout point. I see a lot of skiers standing lazy and then hoping they can magically turn the ski for the pull out and achieve a good body position simultaneously... Narrator: "they did not"
If you are standing well on the ski and set before you pull out then it is much easier to be standing well on the ski when you pullout, glide, and turn in.
2) pull out longer and slower
If I pull out long and slow I have time to watch the gate as I'm coming up on it and make adjustments to timing to compensate for wind, water, etc. If you give your brain time to do the math, it is really really good at doing the math
If you pull out with a "hit and glide" mentality then you just have to hope that you happen to be in the right place, at the right speed, at the right time... good luck
3) Keep my handle close (and a little ahead)
It is important to feel tension on the line throughout the pull and in the glide. If you lose tension in the glide then by definition your gate turn will be turning into slack line which will always be less effective than turning into tight line. This is much easier if you are adhering to the #2 above because it is easier to control tension when you are moving in and out of your pull in a controlled fashion than if you go from standing, hit hard, glide in jerky motions. If you transition smoothly through the pullout it is easier to maintain tension.
(and a little ahead)
In the glide I try to keep the handle at near-hip height but about 4-6 inches in front of me and in-line or slightly inside my right hip.
When you are pulling, your handle is between you and the boat, it might as well start there. If it starts beside you then you will have to fall away from it before you can pull because it needs to get between you and the boat eventually.
Putting it together,
After getting up be sure to get set into good body position pre-pullout.
Pull out longer and slower so your brain has time to do the math to put you in the right place to turn in.
Keep tension on the line throughout the pullout and glide with the handle leading you slightly
Turn into tight line with good connection to the handle.
I make no assumptions to what the pros are thinking about but I've noticed that they do seem to be standing in good position pre-pullout, their pullouts are smooth and connected, and their handles tend to be leading their bodies a little as they come up on the gate.