A good racer, who can handle speed, ice, and put the skis exactly where and how he wants, is going to be a better skier than someone who doesn't. "training" is exactly that: it trains your body to perform at a high level, vs. just going out and skiing. Big-mountain training (like the Squaw Freeride team) is similar: they may not be running gates, but I guarantee you they are doing technique skill building, learning how to flow and attack the mountain, and how and where to turn, not unlike racing. Whether race training specifically is required: probably not, although it is no coincidence that most of the world's best skiers come from race backgrounds. What is required is skill building and tactics, not unlike any other sport you might name. It pretty much goes without saying that someone who races on the World Cup probably can handle most terrain out there, with ease. Not that it is required to be a top level skier, but certainly would be helpful. Any high level skier will have spent thousands of hours perfecting their technique.
Also, the term "good skiers" is relative. I know people who can survive terrain on the hill, but pretty much suck and can barely buy a turn. Since they ski fast, they are considered by some to be "good skiers". The people I am talking about are good skiers: they can flow and be smooth on the scariest terrain, not just get down alive. There are very few skiers like that around.
On a side note: what is with the anti-racer mentality, anyway? Every one of the good ex-racers I know (who raced as juniors) are some of the very best skiers on the hill. Are some of these people offended that they can't ski as well as a person who grew up racing as a kid? I remember last year skiing with a bunch of bro-bra types at a Head demo, testing out the freeride line (I was on the 110mm model, they were all on something similar), and proceeded to get dropped in short order (in deep snow, nonetheless) buy an ex-junior racer buddy of mine, on his own Sollie Fury's. He might not have had the most ideal skis for the day, but he rips, so it hardly mattered.