Each player would simply keep a rook on the back rank negating the "touchdown" and as for the rule of repitition...which player is forced to make a different move? By arbitrarily choosing one, you give the advantage to the other.
Interesting Idea for Chess Variant
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A rook on the back rank can easily be forced off by other pieces, blocked, or distracted with the defense of his king. And you're forgetting about endgames with rooks traded off. As for the the rule of no repetitions, the player who would be repeating the position for the 3rd time would be forced to choose something else, as he or she would be the one typically forcing the repetitions.
I recently thought of one simple idea that could make games a lot more interesting. Have two ways to win. Either checkmate your opponent's king, or advance your own king to the opponents first rank (in the case of white obviously the 8th rank). This would make for very interesting play, as not only would you have to consider mating possibilities, but guard against the opponents king going for a "touchdown" and winning the game.
Additionally, if one wished to eliminate draws altogether, the rule of repetition could be eliminated (force opponent to make a different move), and stalemates would result in either a win or a loss for the stalemater (I think a loss for a stalemater would be interesting).
What kind of effects do you think this would have on play? Would it lead to a lot more chaotic endgames? What does everyone think?