I'm really not understanding this hard line you're drawing. The jumping rules determine what a character can do without possibility of failure. I.e. they automatically succeed, no check required. There then has to be a certain amount of reasonable distance beyond that the character can attempt with some risk of failure. And a further distance at which the character is guaranteed to fail.
The character stating that they realize they're attempting to jump beyond what they can do without fail (but not an unreasonable distance) puts them into this grey area surely?
I don't get why you're being so black and white about it?
The only thing that's black and white is the need for the player to describe a goal and approach. The gray area is how the DM judges what is offered. I would not accept something along the lines of "You know, I just try harder..." as a viable approach to the goal of jumping an unusually long distance. But some other circumstance may be viable such as taking advantage of a terrain feature or resource the character has. You might be okay with "You know, I just try harder..." It's not a call I would make, but neither of us are wrong.