I've made the same point before, though with a somewhat different conclusion. The usual point here is "we should run spells through the skill system somehow, so everyone is on a similar playing field," which I think is exactly backwards. There's two things that are at play here.
Spells are a better game: Assigning limited resources to overcome obstacles is a more interesting gameplay loop than rolling dice for a chance the problem is resolved. Spells are proactive, in that you have to decide to use them and consistent. They let players set the terms of an engagement, both by deciding what problems their resources are best spent on, and by allowing players to shape the board before a given obstacle even arises.
If your goal was just to build an exploration game from scratch and you could only have a spell or skill system, your game would be more engaging if you went for spells. Skills aren't generally a player deployed mechanic, they're a reactive defaulting system. You're not using a Stealth check or a Persuasion check, you're trying an action, and if you don't have a resource you can expend to make it happen, you default to rolling against a % chance of success.
Mundanity is defined by interaction with default systems: Here you get to the aesthetic problem. We've come to define magic as exceptions to the defaulting system, and mundanity as using the defaulting system, and the association is incredibly hard to break. Normal "skill" has to be slotted into the same portion of your rules that defines actions taken without expenditure, and use the same % chance to activate resolution, or it is no longer perceived as a function of skill.
With that in mind, I think the best way to express mundane skill while allowing access to the more engaging resource expenditure game is a two-pronged design. You start by putting more abilities into the defaulting system explicitly. Make it clear what a character can climb with what check results, and then ensure you have more powerful options at the higher end
outside the RNG for the range in which they're level appropriate. That is, when characters are expected to have +3-5 mods, set a DC 25 ability to move full speed while climbing, or sense emotions so well you get surface thoughts or whatever.
Then provide your mundane characters with skill modifying abilities in their character class. Rogues can use a pool of focus points to modify their checks, or fighters can exert themselves X/times per day, etc. You can also add in character class specific abilities you want, as additional skill uses you unlock with a class ability (the classic example being a rogue specific ability to open magical locks, perhaps taken to the more useful "dispel magic through interaction") or perhaps through allowing skill swaps, so actions can be taken with a different skill than normal.
Fundamentally, the game is better off if players have abilities that do things and those abilities have an associated cost. The game then becomes about deploying those resources at the right times, to greatest effect. Then, when characters don't use resources and to differentiate characters with exceptional abilities (which I would argue, should be all of them) you do want the default resolution system provided by skills to tell you what happens when they aren't spending resources to conqueror obstacles.
Edit:
Here's something I wrote about this a few years ago elsewhere that I think neatly summarizes the point.
The tl;dr is that we should probably demote skills to the thing you only use when you're not spending class abilities, and if the rogue/fighter can't survive that, we should cheat and put some portion of their class abilities in the skill system.
