In higher leagues you'll need to play 100s of games every day for the whole week to advance... so "saving" a rating doesn't really work, but sandbagging would work. There are other ways to abuse it too, for example having multiple people play on the same account would let your account play 24 hours a day.
But sure, it does make certain time controls more popular. 5 minute chess and 3 minute chess are both blitz, so both earn 9 trophies per win, but obviously you can play more 3 minute games than 5. And since arenas give bonus trophies, it also encourages arena play.
As for it mainly encourages more playing, yeah. Being in a high league means there was a week where you spent a lot of hours playing. It's a bit like a "best win" statistic except for hours played in 1 week.
So I was playing chess games and discovered I am number one in my league by a mile. That isn't to brag or anything, I wasn't playing better than anyone, I was just playing more.
It seems to me that leagues just encourage people to mindlessly play game after game. You don't lose points for losing, so even if you win just 30% of your games, you can become #1 if you just play a lot. To my knowledge, I haven't played with anyone in my league, even though we are the same rating, so losing does not benefit my competitors.
I have also noticed that the scoring system disproportionally seems to benefit certain formats over the others. a 5 minute game has a prize of 9 points for winning while a 10 minute game has 15 points. In the time it takes to win a 10 minute game and win 15 points I could invest the same time into two 5 minute games and win 18 points.
You would think this system would encourage people to try out different time formats but I am here to tell you that it actually has a negative effect: people are encouraged NOT to play other formats until they advance into the higher leagues. By staying low rated in those formats, they can save them until they get into a higher league where the point race is more intense, and then finally play in different formats where they are vastly better than anyone else for 20 games.
It isn't fair to players with a balanced rating across the board. Even if someone comes anywhere close to me, I could just switch from rapid games to bullet games (which I am way lower rated in because I haven't touched), win 20 games in half an hour, and pull ahead again.
So I ask, what is the point of the league system, when if we assume that players should want to win their league, it encourages them not to analyse and learn, but to immediately hit the 'next game' again and again, and it encourages players to only play one format in order to 'save' them for higher league levels, where they then use it to gain an unfair advantage over others?