So, this is what I've decided I am going to pitch to our group on Saturday:
* Proficiency caps at +8 (no change for us)
* Ability caps at +4 (18 is the max, regardless of racial modifiers)
* Expertise grants advantage on skill checks for selected skills. (We stack sources of advantage and disadvantage, so this still makes things like Elven Accuracy and certain magical items valuable).
* For attack rolls, saving throws, and skill checks, a natural 1 on the d20 always fails and a natural 20 always succeeds.
Class-wise:
* Bards will begin with 4 skills, not 3, for their class (not that it really matters to us, but just in case we ever have a bard again...

)
* Rogues will be allowed to select 2 additional skills for expertise at 11th level.
* Bards will be allowed to select 2 additional skills for expertise at 16th level.
I am adding a new Feat:
Raising the Bar
You have ability beyond the norm. Select one ability score and you gain the following benefits:
* You gain 1 point to the chosen ability score.
* Your maximum for the chosen ability score is now 20 instead of 18.
* Once per long rest when you make an attack roll, saving throw, or skill check using your chosen ability score modifier and the result fails, you can reroll the attempt. (MAYBE??? Too much? I probably won't keep this one... but our long rests are 24 hours.)
I am also doing a re-write of the Prodigy feat:
Prodigy
You have a knack for learning new things and a natural talent in one skill. You gain the following benefits:
* You gain one skill proficiency of your choice and proficiency in one language of your choice. You also gain proficiency in one kit, tool, or vehicle-type (land or sea) of your choice.
* Choose one skill, kit, tool, or vehicle-type in which you have proficiency. Whenever you make a skill check using your chosen skill, you add double your ability score modifier to your check (minimum +2).
I like the change in prodigy to base it on ability score modifier instead of proficiency, since a prodigy is supposed to have a natural talent bend IMO. Also, anyone can be better with a selected skill, etc., but no one gets it for free as a class feature. All other sources (such as Knowledge Domain's "Blessings of Knowledge" for clerics) that double proficiency bonus will grant advantage instead.
This means, barring magic or some other boost, normal bonuses will not exceed +12, up to +16 via the Prodigy feat (which anyone can take now). If you invest in max ability score, Raising the Bar, and Prodigy, you could have a +18 max. That is a heavy investment, and even with expertise granting advantage has about a 30% to fail at a DC 30 check and even 9% against a DC 25.
For the people who play without feats, this won't work, but we do and I think it will work for us.
Now, there is still the issue of linear bonuses raising the floor. Other than the 1 always fails rule, I can't do much about that without changing the base d20 mechanic, which I am loathe to do. A rogue who is +8 on stealth with expertise against a passive perception of 14 will succeed over 90% of the time, but I doubt anyone else will argue against it. As others have suggested, higher passive DCs, more foes to check against at once, etc. will have to serve to level the playing field some.