I've been working out the specifics for my skill system for a while now. It's really just tweaks on the system, but some thoughts:
I have some situations where you must be trained to attempt a skill check. For other circumstances, if you are not trained, then a failed check is probably just failure, no chance of success. Whereas for a trained individual there are probably other consequences or complications, not just outright failure. Which ties into the next point:
I use a variation of Take 20, in that any task that you can succeed at, you eventually will, given enough time and no other serious consequences. The difference between the DC and your die roll is what sets the time period. So if you're trying to pick a lock before some guards arrive, you know that you can eventually pick it, you're just not sure how long it will take (since you don't know the DC).
Group checks are either by the best or worst in the group, depending on the situation. For example, Perception is probably the best in the group, with advantage to indicate the benefit of more people. Most of the time this is passive Perception. On the other hand, for stealth at the group it's the worst score, possibly with advantage if there's a way that the party could explain how the rest of the group is helping them. In most cases this is also a passive check.
I use passive checks a lot, this includes for times where they are skilled and the passive skill is too low, but they have the ability to succeed and there's no compelling reason to work through the checks. I call for an actual skill check only when the stakes are clear and it's necessary. For example, while exploring a cavern, I don't typically call for climbing checks if they are capable of success, trained, helping those that aren't trained, and they aren't rushing, etc.
I have a number of specific abilities that are more specialized. For example, one of the characters is a merchant, so they have proficiency in Wisdom (Insight), but expertise in Wisdom (Insight) checks to appraise an item. These are just noted on the sheet, and not a separate skill that's tracked for anybody else.
I add modifiers to skill checks based on your actions and what you tell me. So the more you can describe what you're trying to accomplish and how, the better your chances are likely to be.
I also will frequently use the three successes before three failures mechanic of death saves (I use it for a lot of things). In fact, advantage/disadvantage, death saves, and the exhaustion track are probably the rules I consider almost defining rules in this addition and appropriate them for many other circumstances.
Overall I really like the rules system as a whole, and the current skill system actually fits pretty well. I do allow alternate abilities for specific skills because I think it just makes sense. The old non-weapon proficiency system had far too many choices, and the problem with the increased number of choices is that you either can't pick enough of them, or people end up picking the ones that give them the best benefits, like in combat. We haven't come across a skill we think should be present.
The tool proficiencies don't come up too often, except for the Thief's Tools of course. But I like the concept.
I do allow learning another skill through training, although it takes a long time, and nobody has taken that up (it replaces the Skilled Feat).