Moving into check for a draw as black, why is it illegal? Comparison to Racing Kings I am confused why you cannot move into a check when it can force your opponent to capture your king and expose theirs. In such a case why is there no extra move to result in a draw?

In the "Racing Kings" variant, if white reaches the end of the board, black has an additional move to force a draw if they also reach the end of the board on that move. However, this does not seem to be the case in standard chess which seems to put black at a disadvantage. That is black must "defend" and white seems to have an advantage since it "attacks" from the very beginning of the game.

Here is an example of what I mean:

[fen "8/8/2k5/3R4/2r1PK2/8/8/8 w - - 0 1"]


If it is black to move, why can't they capture the rook (d5)?
This would force white to recapture with the pawn and open up the king to a check from black's rook. Since black started second, it should have the chance to capture the king and force a draw.
Imagine a variation of chess without the rules about check and checkmate, where a player wins simply when he captures his opponent's king. In this variation, Kxd5 loses the game to exd5.

Turns out, that's more or less how real chess works. The objective is to capture the opponent's king. If your king is under attack, you must deal with that threat. If there's no way to remove the threat, then you lose.

The explicit rule that you must end your move not in check (both by eliminating the check if you're in check, and by not moving into check) and the rule that ends the game with checkmate simply ensure that the king is never actually captured. Check can almost be seen as a safety measure for weaker players who might not notice that their king is under attack. Notice that announcing check is not actually in the rules, and grandmasters never do it. The rule is effectively unnecessary for experts. And checkmate is sort of like the game forcing a resignation when the result is inevitable, much like how experts will resign when they see that their opponent has forced mate.

If we think of the rules of check this way, then it's clear why Kxd5 is illegal: your opponent would take your king, and the fact that his own king becomes exposed is irrelevant because the game is over.