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Carlos, that sounds pretty weird to me. I think the spirit of the rules of Chess is that you really are trying to capture the enemy king, we just forbid moving into check to prevent games ending prematurely due to a dumb mistake; in variants, unless the intent is clearly otherwise, I think we ought to follow that spirit, which means "check" must be construed to mean "your opponent could capture your king next move". Playing on because white doesn't "perceive" the danger, and then forbidding black from actually making the capture because "the goal is to checkmate", seems to me to fly directly in the face of that spirit.

There's no reason you can't make a game that works like that, but inferring that as another designer's intent seems extremely implausible.

...

I think I see a potential for paradox in these rules. Suppose white pawn d3, black pawn d5. At first glance, both pawns seem to threaten e4. But that means that e4 doesn't exist for either player, which means neither pawn can move there. But that means that it isn't threatened, so it does exist...

Or you look at one side at a time, and say, for example, that white is threatening d4, so black can't go there, so black doesn't threaten it. Then there's no contradiction...except that you could equally well rule that black is threatening it and so white isn't! How do you decide between them?