1) Skills need to be defined as a specific thing, not treated as an afterthought of an ability check.
Agreed. I know in 5e 'skill checks' are just a subsection of 'ability checks' but I'd prefer to get rid of ability checks and let 99% of rolls reside in either Skill, Attack, or Saving Throws. If we need to alter the list of skills to make this happen I am all for it. I rarely call for ability checks and when I do I always think that it didn't have to be this way.
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 with this in principle. I know we can already mix-and-match skills and ability scores but that's not how it plays out most of the time. Especially for players that only have a player side view of things. My problem is that the disconnect messes with the simplicity of determining my modifier for a skill check, writing it down, and then referring to it when I'm asked to make a roll using that check. It may not be that complex in the moment but over time that's a lot of extra work that the current system doesn't require from you.
3) there should be a penalty (disadvantage? -2?) for using any skill untrained.
I'm flip-flopping here. Using an untrained skill should lead to poor results most of the time. That I'm totally on board with. But 5e doesn't balance between carrot and stick most of the time. Instead, it is carrot or no carrot. If you really wanted to widen the gap between untrained and trained you'd get the same result by making proficiency start at 4 and climb to 8.
That might screw with the game's math a bit too much though. Instead I suggest you cap their roll at their ability score for an untrained skill. Make it a minimum of 10 to prevent negative numbers from being a double whammy. Or maybe cap it at 15 and don't bother with checking ability scores. They'll be able to succeed on easy and medium tasks but more difficult tasks would be out of their reach.
4) Skills need to inherently outweigh ability scores on the d20 roll.
I agree. I posted this in the thread with your poll but I would drop expertise to 1.5x and make mastery the new 2x. With a little bit more granularity we can start giving out expertise, mastery, and half-proficiency (a very underutilized mechanic) to more skills. The new expertise adds 3, 4, 6, 7, 9 over levels 1-20. Unless you rolled really well expertise will have your training in a skill outstripping your ability modifiers in everything but your highest ability. There's more to this idea obviously but I like that it keeps everything within the same range that currently exists, mathematically.
5) the skill list needs to be expanded and more skill points need to be provided to all characters.
I already covered this in points 1 and 4 but yes I agree. We may have different takes on how that happens, but that's for another time.
6) Backgrounds should determine how many base skill points you get, modified by class.
It would certainly make backgrounds more important mechanically than they currently are.
7) Tool proficiencies need to go away and having or not having the right tools should modify skill checks.
I can't figure out how to write what I'm thinking on this one. When you use a tool to aid in a skill you should get some kind of bonus. Advantage, a flat -5 to DC, take 10, or something else. But when you craft something with a tool I think that should have some kind of gating aspect. I may edit this later if I can think of a way to say what I mean.