I understand what you are saying, but there are a few reasons why it works well the way it is currently set up. First, while your rating is moving either up or down so isn't your opponents. If you are like most of us playing on this site you probably have multiple games going at once, the laws of averages will take over and you will gain or lose about the same amount of rating points.
The games on this site take weeks and sometimes months to play out to completion, so over that time your play could improve thus wiser to use a rating that reflects your current improvement streak, conversely this would work if you are on a losing streak.
Here is a little math to think about, I am going to skew the numbers a bit for simplicity. Lets say a player has a 1500 rating, goes on a losing streak and ends up at 1200. Now you start 6 new games, against opponents with a rating around 1500, at your new rating where each win gains you 100 rating points. So by going by what you are suggesting you would gain 600 points for 6 wins. That would put this plater at a 1800 rating, even though the players rating was 1500 before the bad spell of games. Because the rating points get adjusted after each win those same six wins might now bring the player up to a little over 1500.
The way that this sight does it is to keep ratings from either dropping to drastically if you are losing or going to far through the roof if you are winning.
I hope I presented my thoughts clearly.
Just before I started the first chess.com tournament, I had a run of bad games which dropped my rating considerably. I lost something like five games in a row. While my rating was this low, I started a game with someone that I play often, and his rating (at this point) was considerablly higher than mine.
After we started the game (making about a move or so a day) I started in the tournament. While in the tourney, I won several games in a row boosting my score, but the earlier mentioned game was still going.
I ended up winning that game, but my rating was augmented based on my new score, not the one that I started the game with.
It seems to me that one's rating manipulation should be based on my rating at the beginning of the game...but that's just my humble opinion.