The purpose of the rules

[Satinavian]

I don't necessarily think that DC setting has to be attached to the GM role, the classical GM roll is just one of many ways to split roles at the table. However if you're using the rest of that role it generally implies that the GM largely controls setting-side material, and the players largely control PC-side material. That roll has two components. The players have defined the PC-side material (skills, what the character is actually doing, equipment), so it pretty naturally falls on the GM to set

Why would consequences of PC actions automatically fall on the setting side of things? It is PCs who are doing stuff, so that natural division would suggest that the player narrates the PCs actions and how they work out.

It could be. If you use that though you inevitably get a pretty poor model which does all sorts of weird stuff at a high level compared to just eyeballing it. That might be a worthwhile design decision if you care less about the model making sense than the players having access to detailed mechanical information, and it's often a good design decision for situations where just eyeballing it is a bit harder (magic, superpowers, and other entirely fictional elements come to mind). There's valid reasons to use it.

You take the length of what an untrained but healthy person could jump and put that down to average attributes and no jumping skill. Then you take the attribute value of an immobile person and set that to jumping distance 0. Then you take what a good athlete could jump and set that to attributes and skills a good athlete should have in your system. Then you do linear extrapolation and have a very easy formula with a*attribute + b*skill. And now you have an easy rule system for jumping distance for normal sized adult humanoid that does not produce any stupid extremes, not even in high level content. If you want you can substitute skill rank with average skill check result to allow some fluctuation but if that is a good idea depends on how exactly your skill checks work.

Honestly, it is easer to do the math for transforming your expectation to proper rules than it is to actually come up with the distances for "healthy untrained adult" or "athlete" which would at least require some googling.