Honestly, I think that the existing Helpful feature in the UA writeup is a pretty good start. Maybe tie it to particular skills, even just ones you've taken Expertise for, instead of leaving it so general, and allow you to 'Help' yourself for tactical advantage as well. I mean, the other way to go would be to define a different mechanical, possibly tactical advantage for each possible skill you could pick or get Expertise for, but that as you said could take a while. The general 'better Help with stuff you're Expert in' concept would be open to lots of interpretation and rulings on specific cases, but considering how true that is for what ability/skill combo you use for particular checks already in 5e, maybe that's as much a feature as a bug.I think that's what OP wants, possibly with a little more of making the skills useful outside of base skill checks. Maybe something pinging off the same design space as the late-3e Knowledge Devotion feature or skill tricks.
I think Ezekiel is right that one of the issues D&D has is that much of the potential design space that could be explored with additional exception-based functionality outside of spells has traditionally been done via spells (classic example being Aragorn's herb lore becoming oD&D/1e's ranger spells). There have been occasional subversions of that trend, from tactical feats to battlemaster maneuvers to Knowledge Devotion to the less magical-interpretations of bardic inspiration. There's no specific reason why it couldn't be added to, but creating a open-ended (you can be any kind of skill expert) system like this that is balanced with the rest of the game is going to be a significant task.
So, turn the UA 1st level feature Helpful (Help action as a bonus action, not limited to particular skills or situations), into let's call it Expert Help - For any skill or task (including tools) with which you are proficient, you may take the Help action as a bonus action on your turn. For an ability check, it will be clear which skills or tools are applicable. For combat/tactical Help, you may take this bonus action Help provided the skill is relevant to the foe or the environment. Maybe something extra for Expertise in the proficiency, like a choice of a 30 ft range or a longer duration during which the one you help gets advantage on the next roll.
At 2nd level, replace Cunning Action with Multitasking - you may make an ability check based on a skill in which you have Expertise as a bonus action. So, this could be Hide as for Cunning Action if you have Expertise in Stealth, or Search if it's Perception, or being able to make a social roll even while you're fighting someone, and so on.
Then the Inspiring Help feature could again be dependent on Expertise with what you're helping them with, and maybe both the advantage and bonus dice could stick around until used.
That's just a rough sketch, but you get the idea. I think at least the bones of a good little system are there in the UA, and could be elaborated on to be even better. Maybe there also needs to be 'Help Yourself' feature in there somewhere where you can use a skill with Expertise to give yourself combat advantage, not just others, not sure which level it should come in at.