The better team doesn't always win
The best teams in hockey in the 2020-1 season were the Colorado Avalanche and the Vegas Golden Knights, with 82 points. They won 69.6% and 71.4% of their games respectively. So even if you knew a priori that they were the best teams and picked them to win, you would only get about 70%.
The worst team was the Buffalo Sabres, with 37 points. They lost 73.2% of their games.
Not every team wins or loses 70% of its games
Those examples are the most extreme. But many games do not include any of the Avalanche, Knights, or Sabres. What happens when the New York Rangers face the Montreal Canadiens? Who's the better team? In games like that, your chances of being wrong are much higher.
Not every opponent is average
The Avalanche and the Golden Knights are in the same division. Some of the time, they play each other. Again, your chances of being wrong are much higher.
Look at the playoffs for more examples. In eight first round matches, five undercards won, including two of the number one seeds losing to the number four seeds, despite home rink advantage. In the four second round matches, the lower seed won every one. Including the Montreal Canadiens, who had a losing record in the regular season but went to the finals. Overall, the better seed only won five of fifteen matches. Of course, the year before, the better seed won a majority of the matches.
It's as useful to us for a team to be at the bottom of the rankings as at the top. Either way, it tells us how that team is likely to do in games. So rather than think of the Sabres as winning only 26.8% of their games, think of it as their opponent wins 73.2%. If we think of every team like that, the Sabres were the most decisive team. If we guessed that they'd lose, we'd be right 73.2% of the time. The worst team would be a team that won and lost equal numbers of games. Such a team would only be 50% decisive.
Give every team a decision score. They will all be between 50% and 73.2% (in the 2020-1 season). If you average those numbers together, you get that the average team is around 60% or so decisive. Hey, that's about the same as the predictions!
So what is this telling you? The fundamental problem with consistently guessing the winner with more than 60% accuracy is that games are only 60% accurate at determining who is the better team. If we had a perfect system that could tell who was the better team, it would still only be 60% accurate at predicting games. Because on average the games themselves are only 60% accurate. This should not be terribly surprising. It is after all, the reason why playoff matches are seven games and not just single games.
It would be better for this analysis to look at head-to-head matchups rather than overall records. But I could easily find the overall records (thanks Wikipedia) and could not as easily find the head-to-head results. Also, I doubt that there would be enough data for a truly rigorous examination. Note that 31 teams played 56 games each. That's less than 2 games per matchup. Even in a full 82 game season, that's less than 3 games per matchup average with 2 (other conference) to 4 (same division, maybe) being the actual numbers.
Neither 2 nor 4 is a large enough sample. Really, 82 is rather on the small side. Statistics usually work with hundreds if not thousands. But that to helps explain the problem. There isn't enough data to make really accurate predictions.
One of the problems with your initial analysis is that I strongly suspect that you are looking mostly at the easy matchups. The Avalanche or the Golden Knights will usually beat the Sabres (at least as the teams were in 2020-1). But other teams are going to be far more evenly matched. For example, in the playoffs, they were 4-2 (so the better team only won 67% of the time--assuming the Golden Knights were the better team; perhaps the better team only won 33% of the time).
If you get to pick which matchups, you can do better than 60%. But if you have to pick every game, then you have to pick a lot of games where either team could win. And naturally the matchups that you measure are those where you came to a decision. If you couldn't choose between two evenly matched teams, you don't score yourself with a miss. You just figure that you didn't have time. But those hard matchups should pull down your percentage.
This is why the comments keep telling you to try it. Because if you actually tried to pick all the games this season, it is very unlikely that you would do much better than 60% overall. You might well do better than that for some teams though.
It's not clear to me that even a perfect predictor would have a practical use. What does it change for managers and coaches? Anything short of perfect, they still have to play the games. Perhaps it might help them plan a bit as to when to rest players (if you're going to lose, might as well lose with your second team on the ice). But even then, what if that would have been the game where you would have beat the odds?
For bettors, to have a bet, you need two sides. So the weighted win percentage in betting is always 50%. It may feel like you're betting against the bookie, but that usually isn't true. Bookies don't normally pick a side. Instead, they lay off their bets with other bookies such that, they come out a little ahead regardless of what happens. If there was a service that provided perfect predictions, then everyone would use it and you could never place a bet. There would be no one to bet the other side.
With our current imperfect system, what happens is that they use odds and/or the spread to balance things. But the average return is such that if you bet both sides, you will lose money. Because the winnings are less than the cost of placing the bet both ways so that the bookie has a profit margin.
That's not to say that you can't win individual bets. Or even beat the odds on average. But everyone can't do that all of the time. Someone has to take the losing side. People who tend to beat the odds are the ones who pick games where bettors are likely to be irrational. For example, where people are betting on their home team, even when it's not that good. That can skew the odds.