This is an interesting idea. As far as I know, this approach to mitigate white's first move advantage is new.
Unfortunately, playing for king capture removes stalemate from the game. Stalemate is nice because it gives the losing player something to play for.
You could, instead, address it like this... If white inflicts checkmate, black may make one more move if and only if it results in checkmating white. In this case the game is a draw.
Now it gets complicated. In orthodox chess, the purpose of the rules regarding check and checkmate is to ensure a "proper" end to the game, where one player inflicts on the other the condition in which he cannot get his king out of check, a condition in which the king cannot avoid being captured if that were a legal move. Under those rules, a player cannot blunder into a loss by leaving his king en prise, because doing so would be an illegal move. Therefore, it is illegal for a player to leave his king in check or to move it into check, raising the possibility of stalemate, a condition in which a player has no legal move. With careful play, stalemate can be avoided, so stalemating the opponent is a blunder. Ironically, if capturing the king were the object of the game, the condition known as stalemate would be zugzwang, in which a player would have no better move than to expose his king to attack. In that case, stalemate is not a blunder but a desirable condition to inflict on the opponent. So while the rules of check and checkmate are to prevent a player blundering his king away, those same rules make it possible for a player who has a winning position to blunder into a draw. In my view, the zugzwang interpretation is more logical than the stalemate interpretation of the position in which a player has no legal move.
You proposed the rule "If white inflicts checkmate, black may make one more move if and only if it results in checkmating white. In this case the game is a draw." The problem with that rule is that it does not allow black to respond to checkmate by putting the white king in check. Under the existing rules of Even Chess, black doing that leaves white with two options:
1. White captures the black king, black captures the white king, and the game is a draw.
2. White gets his king out of check, leaving the black king in checkmate.
After option #2, black can still check the white king. Theoretically, the game could go on for several more moves while black is in a checkmated position, by continuing to put the white king in check. At some point, black runs out of checks, in which case he loses on the next move when white captures his king and black cannot capture the white king, or he succeeds in checkmating the white king. In the latter case, white captures the black king on the next move, black captures the white king, and the game is a draw.
Still, stalemate can be preserved in Even Chess by adding a somewhat arbitrary rule:
6. If a player whose king is not in check cannot move without putting his king in check, the game is a draw.
As I wrote in the first paragraph above, I believe the zugzwang interpretation of the stalemate position is more logical, and it is consistent with rule #4, that a player may move his king into check. However, in the spirit of collaboration, I leave it up to members of this forum to decide whether to add proposed rule #6 to the rules of Even Chess. Another possibility is that rule #6 could be optional. Eventually, the majority opinion will be respected.