I totally support this, too!
One major part of playing competitive is to get some kind of proper feedback on how well you are doing compared to others. As pointed out by Iguana, the current EAS system and the position in the rankings do not reflect this well. The result of just one game can make a difference of ~30-40 points. If you have two games in a row with bad luck, you are down 50-60 points and there was nothing you could do about it. How am I supposed to care for my ranking at all? Determining the overall winner is ok with the current system but only a tiny fraction of players is able to get there. Most players probably want to have some reasonable ranking for themselves, which the current system does not provide in a good way.
I played a lot Arena games over the year but currently, I totally lost interest because of the very reason stated above.
And as stated before, once you hit a very good score, you are kind of encouraged to stop playing. The higher my score, the more stressful I feel during games. If we instead chose the peak score for determination of rankings, then each time you hit a new high score you would feel very accomplished in a meaningful way. Hunting for a new highscore (while your current highscore is not taken away from you) is much more positive and fun than the reliable stream of frustration induced by the current system.
I don't see any problem in the presentation/display of two columns in the leaderboard. "Current Score" and "Max Score" are categories that are really easy to get. What could be a problem with max score is that your current game result may not impact your ranking immediately. This could actually keep players away from Arena or be perceived as less fun than a system that allows the very next game to always have an immediate impact on your ranking. Also cheating could be encouraged. If you can loose repeatedly without impact on your rank, this might open the door for some nasty shenanigans.
As a solution, we could apply a weighted average of "Current Score" and "Max Score" (e.g. 50/50 or 25/75) as the best reference for rankings. (This has also been suggested before.) IMO this would allow us to have the best of both worlds. But overall, simply taking the peak score would already definitly be better than the current system.
(If you don't, I will eventually.) 
