I've noticed that it's allowed to pretty much ignore the 2nd Player in 2 Player Mode (I.E. Allowing the player to use 1 Joy Con) and it overall turns 2 Player Mode into a slower 1 Player Mode. For me 2 Player Mode should be changed to 2 categories, a Single Player 2 Player run, and a 2 Player Co-op run; and turn 2 Player Co-op into the main category. What would you think of this?

Nothing is being ignored, the rules for 2P state that 2P mode must be turned on, there are no exceptions to whether someone does 1P2C or 2P2C, adding these exceptions have been agreed upon by the mods to be not the best choice, as the category itself is inactive/unexplored, 1P2C will never be faster than 2P2C regardless, so I think there is no good reason to have any form of exception of 2P2C only or splitting the two in any way. You can easily tell which runs are 1P2C and which runs are 2P2C, filtering the category towards these 2 ideas have been noted to have no effect, as like I said, you can tell which runs are which, and are easily comparable right from the leaderboard.