I think instead of 2 endorsements wherever you like, it should be 1 endorsement available per team.
So, if players were thinking about who they thought was the mvp of their own team AND the enemy team, maybe some more sportsmanship would occur, and less toxicity. People would be thinking positively about how well the enemy is playing and how to counter that. If there is a forced reason to pay attention and recognize the best player on your team AND the enemy team, I think it would also elevate the level of game play.
If people were rewarded (by the endorsement points) for respecting and recognizing the enemy team, they might be a little more focused on the game and less on lashing out at their own teammates…