Problems arise when countering a Skill check with a Ability Roll. Skills are 1d20+ skill, stat checks are 1d20 + stat mod. A huge difference at higher levels in favor of the skill.
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Yup. Rolls should always be matched.
Ability Score vs Ability Score; Skill vs Skill; etc.
Otherwise you get extremely unfair advantages.
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I don't necessarily disagree with the idea in principle, but the problem lies when you don't have a skill that has an opposed skill. It is easy when you have "move silently" vs. "listen", or "hide" vs. "spot", but it falls apart when you are dealing with skills that have no opposing skill. For example, what is the opposing skill for "perform"? You could claim "spot", but then you are stretching the concept of "spot" out of what it normally covers, which is another one of your pet peeves you listed above.
In most cases, you should have skills opposing skills, and abilities opposing abilities, but that is not always the most feasible or the most logical of outcomes. In the case of "perform", while it might be nice to have an opposed skill, we don't have one that fits right. In consequence, an ability roll works well enough. As far as the idea of "unfair advantages" goes, I think this is a red herring -- if someone has an incredibly high perform skill, then there is no reason why their disguise shouldn't be virtually flawless to an observer (think of modern makeup and costuming in movies -- an actor can go in and become completely unrecognizable when the makeup is complete). Additionally, however, a low level player with a high INT or WIS ought to have a chance to see through a relatively poor disguise, regardless of whether they have any points in a skill like "spot". By not having it an opposed "spot" check, you also avoid the use of things like clairaudience/clairvoyance allowing people to penetrate disguises (after all, enhanced spot and listen shouldn't necessarily give any insight into a good disguise).