I have read your post three times and I still can't figure out how a sling is of any assistance at all when the gun is on quad sticks. How does it make your hands any freer to grasp the the sticks or forearm? If sticks are used correctly, the shooter grasps BOTH the sticks AND forearm (unless maybe when shooting a cannon at dangerous game ... and I certainly would not want my flexibility limited front and back in that situation ... not any more than neccessary). For me, a sling just gets in the way of mounting the sticks efficiently. I have to try to hang onto the gun AND sticks while getting the sling out of the way so it's not laying between the forearm and forward crotch of sticks. I don't have three hands. Admittedly, tripod sticks are no easier which is why I try to detach the sling in advance of shooting.
A U-shaped or flat bottom crotch may allow for more lateral movement at the front of quad stickd, but not a lot. I hunt mostly rocky broken ground in South Africa. Quad sticks are a pain to set up in that terrain, especially on a side hill, and a bigger pain to readjust when the gun is on them. A regular tripod can also be a pain to set up in rocky terrain but they require horsing around only one anchor point, not two.
Again, for the novice who's rarely if ever fired a gun let alone pursued live game, quad sticks may be a godsend. Those folks are likely to be very unsteady. They are also less likely to be hunting rugged terrain. Quad sticks may improve the odds of a good hit for someone whose inexperience stacks the odds against them. For those of us who are more experienced, quad sticks are "too much of a good thing." Overkill.