When your dog doesn't square up to you, it's usually because he is anticipating your next cast. In other words, he has his mind set on the direction he wants you to send him - he's second-guessing you. From the judges perspective, the dog will appear to be responding poorly to your casts and you will appear to be a weak handler.
The easiest way to develop a nice square stop/sit is to call the dog in just a few steps and repeat the stop sit until he sits facing you squarely. Another important technique (for training sessions only) is to give him more variation so that he never knows which way your next cast will direct him. Double and triple T patterns with lots of variation will break him of second-guessing your next command.
As for your second paragraph on what judges think, I've never once thought that when watching a dog run. Typically its momentum that keeps a dog from getting turned all the way around. Most of your post seems focused on testing vs training. Big difference
You teach or review the skill in the yard at short distance. When good, which for your pup will be in one session I guess, go do BB blinds at longer distance and more field like setting. When good, add distance and do the BB blinds in another location. When good, go back to the field and run cold blind.
This is the key...
You have taught the dog the required response in the yard, you have transitioned to BB blinds and maintained the standard by correcting for crooked sits and you have praised for good square sits. Now you have the tools and the dog has the understanding to correct the dog in a field setting.
I almost caused myself a real problem trying to square up my dog's whistle sit. I probably did it too much, and it got in his head such that he lost momentum and started what I would call almost popping; he wouldn't turn, but his momentum was terrible and you could tell he was at the ready to hear the whistle and stop.
I gave him about 3 days off and then ran some very short pile stuff for about 3 days with no whistle sits at all. I even went as far as throwing some of the bumpers back to the pile and running him to it as a mark, which caused him to throw dirt on me as he left, and praising the heck out of him when he came back. I very slowly reintroduced the whistle stop, at a much lower frequency, and now he is much better.
We have just started swim-by, and for various reasons I can't do it every day, so on the days I can't do it I am still running him to the pile in my yard. Curiously, after about three sessions of swim-by, the little shopping issue he has always had seems to have cleared itself up.
It still amazes me how these animals figure all this stuff out. I wish I could say I was helping, but sometimes it feels like he may really be doing what he is doing in spite of me. ;-)
I lean toward Howard's take on square sits. I do have a treatment for dogs that are chronic crooked on whistle sits because I believe it affects the overall accuracy of how most dogs cast. But it should be considered that some dogs with a crooked sit do it to such a small degree that it's not worth creating a battle about.
I also agree that it is an issue that sometimes goes away through attrition. Give it some thought before making it more of an issue than it may be.