The problem is that you usually have to run your own master server for a game that might not have players in a few years, if months. And the potential cheaters, which can’t be gotten rid of 100%.
I hate it. Seeing how much better other people are at a game is just depressing. I don't want to compete with other people, I just want to play my game in peace. Also, I don't trust games that require a network connection to not spy on me.
I don’t like high score competitions but I do like when games give you stats about how other players play the game. The puzzle game Catherine would tell you what dialogue options other player’s picked during certain story parts of the game which was cool.