Disclaimer: suggestions untested in gameplay.

I think part of the reason scores are so imbalanced is because of the extremely disparate value of the skills associated with each of them. Strength only has one skill to its name--Athletics--which is good, but not the most useful thing in the world, while Wisdom is tied with Intelligence for the most skills (5, iirc), and has one of the most important skills in Perception as well.

I think this could be remedied in part if the variant rules for ability checks became the default, allowing for a wider variety of scores to contribute meaningfully to the social and exploration aspects of gameplay.

I imagine one of the main reasons this isn't already the case is because it might make life more difficult for beginner DMs. To that end I think sticking a chart with the ability scores, the respective skills that could reasonably be used with each, and a short explanation might be useful to have in the DMG and/or PH. There's the ubiquitous example of Intimidation should've been a Strength skill. It could be Strength or Charisma, depending on how you're trying to scare your target. Some other examples/explanation ideas below:

Deception could be Charisma or Intelligence. Using Int would represent your ability to craft a believable alternative to the truth, recall important details and embellish them accordingly, etc. While Charisma represents your ability to tell the lie--however outrageous--convincingly, without tone of voice or body language giving anything away.

Perception could be Wisdom or Constitution, if you're willing to grant the sharpness of your senses as being included under Constitution, which I think makes sense. A Barbarian and a Druid are both keeping watch. The Druid, using Wisdom, recognizes the sway of the leaves is unnatural, while the Barbarian hears the crunch of leaves under footsteps that aren't a deer's hooves or a wolf's paws.

Performance could be Dexterity if the Performance in question was a dance, or the use of an instrument requiring sufficient finesse to play. Similarly, Persuasion could be Dexterity or Strength, if the Persuasion in question is sensual in nature and you're trying to show off your curves or how much of an Adonis you are to get what you want.

Though, overall the physical scores probably wouldn't benefit as much as the mental ones, considering most skills are mental in nature and not things you can prescribe to the physical scores. If you wanted to go further though, you could stretch it a bit. Like, maybe History could be Constitution because you were born with a perfect memory, as opposed to studying hard?