I don't have a problem with advanced stats, but I have a problem with the way many people use them.
First, they represent only the sliver of the whole picture. Take GF%. Player A has a >50% GF%, and player B has <50%. Well looking at that one stat, player A is the superior player. Its obvious, when he is on the ice, goals get scored for his team more than against, and that is the point of hockey. However, what if I told you that player A was playing on a really good team, with 2 great linemates, playing against lesser competition, and getting exclusively offensive opportunities while player B was on a horrible team playing with horrible players, getting buried in the d-zone in the toughest situations against the best the NHL has to offer. Now, who is the better player? Its not so black and white when you add that extra data. Player A could be better, or maybe Player B would be better if he had the same opportunities as A. When testing a hypothesis in a lab, you hold everything else constant and vary one thing. It is impossible to normalize all other factors when comparing players because players play games to prove which team is the best on that given day, not which player is the best overall.
The second problem I have with how people use advanced stats is that they do not say what people think they say. We have no way to measure what we want to measure individually because of all the variables discussed above. So we try to find proxies for those. But the proxies are imperfect. Corsi is used to determine possession, but it is just a proxy. A team could come down with the puck, be effectively funneled to the edges, rip off a poor shot immediately, get a rebound, pass it back and be forced to shoot wide as all lanes were blocked . That exchange took 10 seconds. The other team than takes it down the ice, cycles the puck for 20 seconds, gets a great shot and scores. The CF% for the first team is 67%. The CF% for the second team is 33%. Based solely on Corsi from that exchange, the first team is better on possession and/or defense. But in reality, the second team had the puck twice as long and was more effective in stopping the other team, forcing them into bad shots, while they could take their time and get off one high quality shot.
The third problem is sample size. Stats are aggregates of a lot of data. The more of it you have, the more accurate your data. A player plays 1,000 games, including hundreds of clutch situations, and he's money. Then he plays 1 playoff game, does poorly, and he is a crappy player who chokes when it matters. The larger sample is more important than the smaller. Outliers can ruin a small sample, but get smoothed over with enough data. OR Sobotka was a Goal per game player last year. Is he really a goal per game player, or did he just have a goal in the one game he played.
Those are all extreme examples, yes, but hopefully they show my point.