...you'll never find a satisfying understanding by comparing it to what happens in other sports.

Soccer is its own game, with a totally unique style, flow and sports culture. It's not for everybody, and I can understand why some don't like it. But, no one will ever enjoy it until they get a feel for it on its own - without any comparison to football, basketball, or baseball.

So, Why can't a dismissed player be replaced? The only real answer is: Because it fits.

That's just the way the game has evolved. As the flow and culture of the sport has developed, the rules have developed accordingly, and they compliment each other.

As for the red card itself, referees have always had the authority to dismiss players. The card system was started within the last few decades to more clearly define the rules and make communicating decisions to large audiences easier. Of course the rules have undergone subtle changes and interpretations over time, and if you watch the sport enough, you'll even see small differences in the way certain leagues enforce it when compared to others.

Oh, and you don't necessarily lose that position. You just have to play with a 10-man formation instead of 11. But you can cover (or not cover) any position you want to. You're always required to have a designated goalkeeper, but that's it. And even the goalkeeper can be anywhere on the field he/she wants to be.