I will answer from a different perspective: why Racing Kings (RK) has a rule to allow black a chance to draw, and why the same logic doesn't apply to chess.
What is Racing Kings (RK)?
Background for those unfamiliar with RK: Both sides start with all pieces (no pawns), arranged on the first 2 ranks of the chessboard, white on the right, black on the left. The win condition is to get the kings from their initial squares on h2/a2 to any square on the 8th rank. Checks are forbidden --- no move is allowed to leave either king in check.
The rules as stated above, favour white heavily. Having one tempo is critically important when a game can be as short as six moves. In order to negate some of this advantage, if white moves the king to the 8th rank and black can do the same immediately, the game is drawn.
How serious is white's advantage in RK?
Even with the added draw rule for black, RK still favours white. In practical play on lichess, eyeballing the opening explorer tells us white still wins more than black overall, even limiting the games to top players (2000+ site RK rating). The imbalance is around 48% to 41% with about 11% draws. This empirically shows white's advantage, and it would be even more lopsided without the draw rule. (And if there were long time control games, the draw percentage would be much, much higher. Such is internet blitz.)
Aside from game statistics, further evidence of the imbalance is the opening theory. 1. Kg3 lines have been very deeply analysed by the community and by Stockfish, and the outcome is that black can just about hold a draw with precise play (the theory can run 20 moves deep at times). The other best first moves 1. Bd4, 1. Nxc2 and 1. Kh3 are slightly less theoreticised, but deep engine analysis and practical play strongly suggests they are drawn too (again, with precise play).
So even with the draw rule in RK, black is still just holding on.
Should regular chess have this rule?
At the top level of regular chess, we have a far higher percentage of draws already. It is also clearly nowhere near as imbalanced as the "raw" form of RK without the draw rule. For balance reasons, the rule proposed here (allowing black to move into check in certain circumstances) should not be introduced to regular chess.