I seem to remember that one of the early thoughts about the new skill system was not that you would simply get a bonus to a roll, but having a skill gave you new options others did not.
If you would go this route, and skills like Diplomacy or Intimidate gave you "tricks" that others simply can't use regardless of how big your Diplomacy is, then the mechancis would basically do what you do in skill challenges - give a reason for the low-modifier individual to do his things.
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I am not saying the above rules are good - especially since they still focus mostly on reducing difficulty class.
Something else could be
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I think this is an interesting way of going. I think I like the second approach better, which focuses on something other than toying with the numbers.
In another thread on skills (I can't remember which one now) I made the comment that Backgrounds seem a bit underdeveloped at present. In particular, nothing seems to have been done to incorporate those features like "Knight's Station", "Researcher", etc into the action resolution system.
One idea I came up with was that these features could be used by the player to establish certain things as already given in play, which other player's have to expend resources and go through actual play to achieve. For example, in play some question comes up - "Do we know the answer to this ancient riddle?" A player whose PC is a Researcher can make a check, because his/her background already establishes that s/he hung out in libraries in the past; whereas other PCs can't make a check until they actually find a library and start reading.
You'd have to think about how to extend this to other features. Maybe a Knight already knows some nobles and squires, whereas other PCs have to actually meet and befriend them in play.
Anyway, I think your second approach is another way of doing this - of using these non-numerical aspects of backgrounds to unlock options - without having to make the skills themselves carry too much mechanical weight. And it can help make a background more than just a skill parcel.
So your first Diplomacy suggestion, for example, we instead grant to a Knight who is dealing with other knights and nobles. While your second Diplomacy example might be available to someone with the Soldier background (admittedly it doesn't hang directly of Endurance, but something can be tweaked to make this fit). A Researcher would have a guaranteed minimum floor on certain knowledge checks. Etc.
You might need to put some sort of resource mechanic into play to regulate abuse (no more than once per session would be the default, maybe).