I never have any wrap at all on my strings - I line the hole up with the nut so the string straight through the hole, pull taught and tighten the screw to lock down the string then tune up to pitch. I have literally no string wrapping - the amount I need to tune up literally means the hole is no more than 90 degrees to the nut so its only touching the post on the edge of the hole. I have heard of those that do drop wrapping a bit more round the post so they can slacken the string down for drop tuning but its really unnecessary to have any wrap at all.
The majority of the 'force' at the tuner is on the edge of the hole and its not a 'straight' pull on the string where it is locked in by the Screw. Turning the tuner so the hole is not in line with nut reduces the force at the screw because its not in line anymore. I hope that makes sense - most of the force on the string is now pulling on the post, the edge of the hole. Its much easier to pull something through a hole if the hole and cord/cable/string/rope etc is in line with the hole but turn the hole (or change the angle of the cord so its not in line with the hole) and you need much more effort and pulling force to over come the angle of force and friction where the cord is touching the edge of the hole.
As for tightening the screw, Its literally finger tight using a coin rather than really aggressive and potentially cutting into the string creating a weak spot that breaks. The locking mechanism hasn't failed me on any of my 5 Core PRS guitars, all of which have the Phase 3 Locking tuners and not one has any 'wrap' at all on the string. 3 of mine do have the floating Gen 3 trem bridge too and the other has the wrap around Piezo bridge so the only one with the Tune-O-Matic type bridge is my 594. All my strings (apart from the Hollowbody with the Piezo bridge that has 11's with a wound G) are 10's and either D'addrio NYXL's or Ernie Ball Paradigms (although I do think I will be replacing my strings with All Nickel). Whether that makes a difference in the amount of stretching I need to do or the tuning stability, I don't know but I can only comment about my own experience and string gauge/type I use.
I have never had a string pull out and the locking screw has always been enough for me. i have had my 594 for a few years now and I have never had any issue with either the tuner or tuning stability once the strings have settled. I rarely have to do more than one pass (that's up and down the string) and a good hard strum test - at most, the strum test may need a small tweak and then repeat the strum test to see if it now stays in tune and if not, a 2nd pass of stretching but I have rarely had to do a 2nd pass and the vast majority of the time, just the stretch is enough because it stays in tune after the first hard strum test. I don't do more than a tone bend, at least very rarely and often go through a set of strings without ever bending more than a tone (maybe a bit more with some vibrato on the tone bend but never a tone and a half bends).
Anyway, I can only comment on my experience with my 594 (as well as my other PRS guitars - although they do have different nuts and bridges) and the string gauge/brand I have used. In any case, I hope you find a solution. I doubt that a bit more wrap (although not a 'full' wrap) would be the issue so maybe try some nut lube and/or try and determine where things are moving or sticking. As for the tuners, you may need to tighten them up (maybe they are moving) and as for 'wrapping', PRS themselves do the pull through taught, tighten up the locking screw and tune up to pitch with no wraps and, if you find pictures of the headstock where you can clearly see the string and tuner, you will generally see no wraps at all and the string is only touching the edge of the hole - not even a quarter wrap.