At least according to the poll I put up recently, Skills are the single most agreed upon mechanic for D&D. Thinking about the way 5E uses skills, I would like to propose some ideas for how I think they could be better utilized in the game.
1) Skills need to be defined as a specific thing, not treated as an afterthought of an ability check.
2) Skills need to be disconnected from ability scores. there is already a rule that makes this possible, but it needs to be explicit.
I agree, although I would say perhaps either/or, take the higher value, perhaps. I endorse the concept that ability score are "talent" where skills are, well, "skill" and tied to level. With Bounded Accuracy I think they need to be added together regardless.
3) there should be a penalty (disadvantage? -2?) for using any skill untrained.
Given BA, I like this, but I think it would be better to gate a minimum competence with proficiency. Right now we have three levels of skill regardless of the numerical value; Untrained, Proficient, Expert. I think we can use that to clarify what actually needs skill checks versus uncanny or preternatural levels of skill.
4) Skills need to inherently outweigh ability scores on the d20 roll.
Yes, but infeasible under BA.
5) the skill list needs to be expanded and more skill points need to be provided to all characters.
6) Backgrounds should determine how many base skill points you get, modified by class.
Good grief, no. If I want a skill based game I'll play RuneQuest (which I enjoy). D&D has classes, which should delineate what they are capable of doing. Each class should have the same number of skills, with a nod towards what they might have learned in an apprenticeship that they abandoned for adventuring (Background). If they want more skill proficiencies, that's what feats are for.
7) Tool proficiencies need to go away and having or not having the right tools should modify skill checks.
Yes, sort of. I think those should be skills, and if there is a tool kit tied to a skill then there is a penalty applied. (Might be the same thing you said.)
In addition, this is a more general rule, but I think advantage and disadvantage should stack from different sources and cancel each other out on a one for one basis...
Yes, to the point of rolling three dice at a time. If you are rolling four dice then you pretty much win, from a Numberphile video I watched once (I think, might be wrong as to the source).
Edit: I would prefer, personally, 20 or less skills. Also, there is also reliable. I don't think that can be applied to a non-proficient skill, but I'm uncertain.