I'll upvote because I understand why you probably feel this way..
Unfortunately as it has been somewhat "rudely" pointed out..
It would create a very negative environment and an every man for himself scenario.
However.. I think we could potentially find some balance..
It really sucks to do extremely good in your role, only to have your team-mates decisions cause a clear victory to become a horrible defeat.
I think winning or losing should determine the majority of your LP gain/loss
I also think that a particular set of metrics for each role could be devised so that players that are successful individually could receive a bit of a reward for their effort.
Assists would increase this score on all roles to deter kill stealing and the fights associated with it, there would be more incentive to give the kill to your carry.
Supports could be graded on their warding, proper pinging (if its trackable), utility, crowd control, and relevant factors
Jungles on objective conversion, assists, dragons/barons/scuttles/rift, low minion cs high jungle cs
Mid/Top/Bot would have very similar metrics with maybe a few specifics thrown in.
This could mean scoring high might give you a few extra LP on a win and softening the blow of a loss.
If you are at the point you are winning/losing 20 LP then a high score could fairly lift that particular win to say 25 LP and a loss might only be 10 that game, where as your 0/14/3 1 Vision Score adc lost the full 20. But you would have to do exceptionally good.. like 10/5/4 24 Vision score with 3 towers and 1 inhib
It isn't much but 4 games of consistently good individual game play wins would be one free match of 20 LP. Losing would be like two of your matches didn't get thrown against you.
It would have to be finely tuned though.. and Riot would have to really think about a system that wasn't abusable for gain. it would also have to encourage players to want to play as a team.
The sodium levels would definitely drop..
Players would relax a little and think.. OK I'm losing this, but at least I won't lose everything.
I've lost games where the fault was on my shoulders.. I've lost games where I practically spoonfed the team a win and it gets thrown over a fight or a lack of focus or skill or something crazy.
I've counted quite a few days where I took a loss, and it felt like my team-mates were nowhere near my skill level.. Games where I've gotten every dragon, taken multiple towers, had double digit vision scores, i'm 4 levels higher than everyone, sitting at near full build I don't take the kills because I'm getting my team fed status.. and then they are fighting over something, divide my vision score and its still higher than all 4 team-mates combined, how the hell are these guys level 30esque stats across the board nothing you do or say can quell the saltstorm, the raging infernos, and you've gone out of your way to talk someone up and its just over
A few of those games it would have been nice for something like that..
But once again.. highly abusable and how do you actually implement it fairly?