I would disagree with you. If you're using a stiff poly especially it is important to push the next string up against the last tensioned string. Which led me to say I doubt the USRSA has been teaching this technique for the last 30 years.
BUT there is a trick to it. If you're using a stiff string it is important you push the side closest to the last clamp up first. If you don't the string wants to slide down away from the last tensioned string because it is so stiff.
The problem is the stiffer the string, the less it wants to bend, and the more likely it is to kink (damage) if you do bend it. It doesn't matter which end you start with, it will still be bending the string the same amount.
Because of this conversation, I realized that my thinking on this began when I was still a very novice stringer. So, I decided to put my thinking to the test today, as I needed to string my racquet anyway. My racquet is a nice and open 16x16 string pattern, so if the string was damaged in this racquet, it most likely would in others as well. I strung it with Solinco Hyper-G 16L, which is on the stiffer side, but not as stiff as the 4G I mentioned earlier.
So here was the test: after I pulled tension and moved the clamp, but before I pulled the last loop of string through the crosses, I pushed the next cross all the way against the cross I just tensioned, which you suggest to make the weaving easier and I do with nylon strings. Then I pulled that section of string out so I could inspect the string to see what condition it was in. It turns out, my thoughts were correct; the string was bent and damaged.
My conclusion: I would never do this on a customer's racquet, because I don't want damaged strings in their racquet. However, at least on my own racquet, I don't mind doing that for the last 3-4 crosses because I know that I'm not going to break strings there, even with slightly damaged strings, as the last few crosses are the only ones that actually get difficult to weave with anyway.
Try it for yourself; with a stiff poly, push the strung but untensioned cross up against the last tensioned cross, but then pull that string out and inspect it. If you wouldn't want string in that condition to start stringing a customer's racquet with, you shouldn't do that to the string as you install it into the frame.