So what DO teams look for in players? What makes a good player good and a great player great? It is my understanding that there have been better players than Mantle and Mays since you can't really compare eras, or is it easier to compare eras in baseball as opposed to other sports?
This is just for hitters, pitching is a bit more complicated and catchers have additional items around handling a staff and pitch framing.
There is some disagreement with this, but I think teams have all realized the traditional stats like batting average, home runs, RBIs, and especially fielding percentage only tell part of the story and are in some cases arbitrary and in all cases involve some luck. Example in one game a better hits the ball really hard 4 times and records 4 outs, player 2 hits 2 seeing eye weak ground balls, a check swing hit, and a blooper that falls and he is 4 for 4. The theory being that over time luck will sort itself out.
More of an emphasis is placed on on base percentage (getting on base via a hit or walk), exit velocity (how hard they hit the ball), launch angle (getting the ball in the air) there are all kinds of new stats with fielding that determine if a ball is hit in a certain area with a fielder positioned in a certain place, and a ball hit a certain way (speed, trajectory, etc) what percentage of the time should an out be recorded. There are multiple metrics for those 4 things, but I think they are the keys most teams look for now.
The whole deal with Moneyball is breaking down what skills and attributes gives you the best chance to score a run on one end and prevent the other team from scoring on the other side.
The more "soft skills" things that are big now are generally being a good teammate and not getting in the news for the wrong reasons. There is less tolerance for assholes who are good players.
The thing with Trout is he does everything really well, and he is a good teammate, and someone the Angels can promote as the "face of the franchise". Which does lead to money both in ratings, tickets sold, and merch sold.
I would add injury history always plays into it as well.
There are different formulas, but the attempt at placing a unifying value on a player is WAR (wins over replacement). The idea being that how many more games will this player win based on having him versus someone that can be picked up cheap on the waiver wire. Combining hitting and fielding.
If you look here, you will see the last 10 years based on how baseball reference calculates it
https://www.baseball-reference.com/leaders/WAR_top_ten.shtml
Last year Trout was 10.2, so essentially Trout accounted for a little over 10 wins on his own. But every years he has played, he has been in the top 10, usually the top 3. Few players can say that.