I don't think anybody's saying you should manage a team just by looking at numbers. Stats are just there to help you make a decision. When you see some number that doesn't match the eye test, you want to find out why. Is that a problem with the stat, is the sample too small, or is there really something worth investigating?
Teams are all looking to gain an edge. Sure, if you can have the best scouts out there, good for you (hmm, how would you know they're the best scouts? you probably look at some set of numbers?), but if you can't you need to find another way.
We're all already using numbers all the time. The whole point of advanced analytics is to go beyond the basic numbers, because we know they're not good enough: W/L, a goalies GAA. We know that run differential is a better predictor of future performance than W/L. We want to find even better predictors if possible. What's wrong with that?
More generally you should always strive to have a way to compare things quantitatively. The trick is to find the right metrics for that. So, sure, some people may jump on the latest fancy stat as if it's a miracle recipe, and then we find out that there's more going on. Well, that's how science progresses. If all you do is just rely on eye test you get stuck.
Baseball has made huge progress thanks to fancy stats, but I feel there's way more variability in numbers in hockey (hence the need for error bars and standard deviations mentioned earlier by someone), and so it's going to be more difficult to find meaningful metrics. That doesn't mean we should give up.