Why Circumcision?

I feel like an idiot to ask this question, but the more I thought about it, the more unusual God's requirement of Abraham seemed.  Why did God choose circumcision as a sign or symbol of covenant between God and man?

First, the male penis houses the plumbing for the exit of waste water from the male human body.  This seems, from a sanitary sense, dirty.

Second, the male penis also houses the plumbing for the transfer of sperm to the female counterpart enabling procreation of the human race.  This seems "dirty" from a whole different angle.

I know, I know, I should not be so concentrated on this topic - but really, why is this covenant symbolized through circumcision?  Why not remove the first joint of the fourth toe?  Or, why not trim off the lobe of the right ear - something visible?

Why would God choose to signify His covenant with man by requiring him to remove a portion of an unclean, unseen, and sexual body part?  What was God thinking?

Father, I know You do nothing randomly.  However, circumcision seems so odd.  I am inclined to give it no thought...  But then, how can I justify giving no thought to a covenant between God and man?  I believe You gave an ongoing message to Abraham that encompassed a lot of "life application".  Give me the grace to comprehend it...