^ Bryce Harper was 19. Mike Trout as well.
Bryce Harper also left high school after his sophomore year to fast track his playing career. He still played over 120 games in the minors before getting the call up.
Mike Trout played nearly 200 minor league games before getting called up, struggling in his time in the majors, batting .220 in about 35-40 games, before breaking out the year after.
Puig has played under 100 games in the minors.
The thing you said is the exact issue with the situation. People see what Trout did and assume Puig can do what he did, but Trout had a historic season not seen in the history of the game. Trout and Harper are fantastic talents, but it doesn't mean Puig would duplicate their success. For every Harper and Trout, there are plenty of prospects who flop in their first go around like a Jackie Bradley Jr. (Red Sox prospect) who tore up spring training then started the season batting .097 and getting dropped back down or prospects who flop, lose confidence, and never pan out.
For the Dodgers, a perfect example of that is Joel Guzman, who they signed out of the Dominican Republic. At 6'6, 225 lbs, scouts projected this power hitting prospect as a Miguel Cabrera or Juan Gonzalez type player. He lasted two years with the Dodgers before disappearing off the face of the earth when they sent him to Tampa Bay.
I'm glad he's finally up with the big club but the decision to keep him in the minors was still a wise one based off history. I didn't envision him getting called up because of injuries, I figured they would trade Ethier sooner than later and bring him up to take his spot in RF. You may be seeing that happen at the moment too.