dark elf could start of in a deep mine.
kahjiit surrounded by a squad of his dead thief friends.
...Why would a dunmer be coming out of a mine? It'd be far more logical for them to be crossing the border from Morrowind
Also, starting the Khajiit surrounded by thief friends presumes that the character will be a thief themselves or are heavily affiliated with thieves and hinders roleplaying to a huge degree.
To people saying that specific entrances for specific races limits roleplaying: How would that limit it any more than the same one for all races? As long as the entrance doesn't really say much about the character in particular it wouldn't limit it. Sure, being in prison is SLIGHTLY less hindersom,e but it presumes you were some form of criminal or at least had to have some conspiracy related to you.
Bottom line: If incorporated correctly this wouldn't hurt the experience. On the other hand, it wouldn't enhance it enough to be worth the effort. Sure, it'd add a slight bit of replayability but hardly enough to warrant coding it all.