One thing they do really well (in this context) is challenge your biases. It's been said many, many times before but as humans we absolutely suck in terms of basing our opinions on something or someone on an anchor point - for defensemen, it's often referred to as "the big mistake". In that sense, our perspective on a player can be skewed based on what might have been 5 seconds in a game, and even into the season. In terms of your "eye test", it's easy for scouts and coaches to scour over video, but even then you're probably looking more at clips instead of the overall game - nevertheless, going over video is likely the best method to help coach the individual to point the specifics of what can be done differently, or better.
In terms of statistics (or lies, damned lies, and statistics), it helps you set benchmarks and key performance indicators (as well as be able to perform some better overall "guess-sumptions" on how nearly 700 players in the NHL are performing in the case of trades). In-game, you can be biased towards the length of time a team spends in a zone (
even though we already know through data extended zone time doesn't always mean more goals for). But, for example, you can review the game and see while there was possession, it wasn't productive possession, meanwhile the opponent has multiple rush chances against you though they haven't scored. Do you keep rolling with that line as they are or do you review video with them to shift them towards a style more conducive to scoring goals?
This is only a quick example. There's a lot more applications, but benchmarking should be a key one. As with any business project, you don't want to go from 0 to 100% completion without being reliably able to say that the project will do what you want it to do at completion? Same thing in terms of using statistics or analytics in hockey.