Alternating would confuse both players and spectators. You'd also have a lot more setup errors since people might forget to alternate. Besides the rule has always been that once you start a game going one direction the whole match must be played that way.
What about alternating the home board after each game?
You can ask for permission to roll on the left side during bear off or you can do something like use a baffle box or dice tube if cocked dice annoy you. A simple thing which many people do not do is to aim the dice so they come out of the cup in an area where there aren't too many checkers. A good spot is to move your dice cup hand from right to left so the dice hit the center barrier of the board and bounce into the center of the board.
So far, this is contrary to what I've seen. A lot of players complain about rolling into the home board, due to more cocked dice.
I and most people are right handed so it is more efficient to have the home board on the right side.