Lefties need not apply. Or why are there two batter's boxes?

We're so stuck in our thinking that we don't even consider just how odd it is that baseball is so asymmetrical.  Aside from the utter absurdity of the non-uniform playing areas, why the heck are there two batter's boxes?

Computer generated picture of Aristotle
By Kolja Mendler Benutzer:Klingsor via Wikimedia Commons

And if there are going to be two, why doesn't the righty batter run to "third" base?

A year ago I did a lot of research on handedness.

1. You really can't switch how you throw.

2. Almost all righty batters throw righty.

3. Only about 42% of lefty batters throw lefty.

Most lefty batters are opportunists, taking advantage of two things:

1. 72% of pitches are thrown by righties, which is good for a lefty batter, bad for a righty batter.

2. Lefty batters are closer to all the bases, not just first base.

Most switch hitters throw righty.

The most significant deviants:

Rickey Henderson: threw left, batted right.  This is especially odd since Rickey crouched at the plate and tried to draw walks so that he could steal bases.  He must have been able to bat lefty, which should have helped him get on base even more.

Lance Berkman: threw left, batted both.  OPS batting righty: .777; lefty .995.  What the heck was Lance doing?  He should have just batted his natural lefty.

There should be only one batter's box, the one for righties.  It's the only accommodation for handedness among the three American team sports.

By practice, the catching position, which should be eliminated, is restricted to righty throwers.  I've never understood this.

Second base, third base and shortstop are also restricted to righty throwers.  This is understandable given that the rules require the batter to run to "first" base.

Lefty throwers are already relegated to the outfield or first base, unless, of course, they are willing to play part time as pitchers.

So why not complete it and eliminate the lefty batter's box?

If you want to make an allowance for lefty throwers, have the batters run to "third" base.  Making third base first base would change the handedness advantage for fielding positions second, third and short.  Only lefties need apply ... but they must bat righty like everyone else.

There, that seems like a logical compromise, which makes baseball much more logical and fair, two bedrocks of America.