Okay.... I googled "A game that doesn't exist" and wound up with "Polybius (square)" which also came up in the commentary below. Hadn't heard of that before.
A game that has existed for over 1500 years" presumably refers to Chess. I thought that might be the 'key' for a simple version of the square or an 8x8 square but it didn't work,...
I found a new solution that is 2 moves better than my previous one!
Big thanks to the answer by Retudin for the new pawn structure idea!
With the new hints this has become much simpler. In fact, OP has (unintentionally?) changed the character of the puzzle. At least some solutions can be step-by-step (like a sudoku for want of a better simile) reconstructed from the hints.
For example number 8 (I'll only do one, so people still have a chance to earn the bounty by solving one of the others):