What effect on flavor and/or cooking does rinsing grits have?
The warning you quoted is of course not a real warning. In the article's context it's read as a cooking advice: 


  Finally, a tip from one of those gracious Southern cooks: "Do not rinse grits, ever!"


What's indeed missing is the reason. The "Because:". Had a southern cook forbidden me to rinse the grits, I would have guessed it's because of the starch. Just as you thought.

Rinsing cleans, moistens the food a bit and removes surface starch when such exists. And many times I've read suggestions against rinsing starchy pieces of food, because that surface starch is "invaluable" (that's the word I've read most).

When heated and then chilled, starch thickens. It thickens soups and sauces, and probably helps with the texture of the recipe that follows the warning: "Creamy-style stone-ground grits". 

When I make a gratin or potatoes with cream, I too guard the surface starch. I've never done a blind taste test to see if non-rinsed potato slices indeed make a creamyer dish. But it sure doesn't hurt.

So I think this is the logic behind the "warning". It just turns out nicely when you don't rinse the grits. And that could be because of the starch. Hard to tell. Hard to tell if those "gracious Southern cooks" know the reason themselves, or if they just inherrited a law whose reason was known only to the guy who didn't make it to the Mayflower. 

So rinse the grits or not. In the south they don't.