Here is another issue that arises: When a player is not reasonably specific as to goal and approach, a lot of space is left for the DM to assume. A common approach that I see in many games including certain very popular actual play videos (and it annoys me) is that the DM then declares what the character is doing because the player was not specific enough. This can often be avoided, as can the the skills overlap issue raised by the OP, by making sure that adjudication happens only after the player has been reasonably specific as to goal and approach. Once that is established clearly, it's easy for the DM to then choose an ability check that applies to resolve uncertainty, and to narrate the result without encroaching upon the player's role of declaring what the character does.
Maybe this is contributing to the confusion.
If a player makes a more general statement, about e.g. examining a corpse for example, then there's room for the DM to interpret the precise details of the examination.
Perhaps in your preferred level of specificity, a player would not make a statement like "I examine the corpse" but rather... "I examine the lividity of the corpse to determine the approximate time of death" or "I examine the contents of the corpse's stomach."
If that's the case, I can more easily see a specific skill taking prominence. But that's just because you've drilled past the level where two skills made sense... and you'll come right back to it, anyway.
At broad, imprecise levels of specificity, a single skill often makes sense as a rough approximation of the action. At more precise levels of specificity, multiple skills may apply. But if you keep drilling down to a single moment of action ("I cut open the corpse's stomach") then you come in a circle back to a single skill.
But at that level of specificity, you could easily need 5, 10, or 20 discrete actions to describe a single procedure... and thus end up with as many skill checks. (Check lividity. Examine stomach contents. Examine fingernails. Examine wound site. Look for hidden wounds, injection sites, etc. And so on.)
So then you're just back to my statement of "sometimes, call for multiple skill checks" except you're calling for them one by one at each step of the way, as the PC provides a highly specific, discrete chunk of action.
Sometimes, that's an exciting series of rolls, I'm sure. But sometimes you want to cover an entire scene with a single adjudication. I think that adjudication could reasonably include 1
or more skill checks.
It's the difference between:
PC: I need to put some distance between me and this crime scene. I'll run across the rooftop.
DM: Make a Dex check, DC 13. Failure will result in you taking a tumble down the steep roof.
PC: Can I include Acrobatics?
DM: Sure.
PC: *rolls* Success! I jump to the next building.
DM: Make a Str check, DC 14. Failure and you fall short.
PC: Can I include Athletics?
DM: Sure.
PC: *rolls* Success! I run across the next rooftop.
DM: Make a Dex check, DC 13. Failure will result in you taking a tumble down the steep roof.
PC: Can I include Acrobatics?
DM: Sure.
PC: *rolls* Success! I jump to the next building.
DM: Make a Str check, DC 14. Failure and you fall short.
PC: *rolls* Success! I run across the next rooftop.
DM: Make a Dex check, DC 13. Failure will result in you taking a tumble down the steep roof.
... and it goes on like that.
Versus:
PC: I need to put some distance between me and this crime scene. I'll take to the rooftops, running as fast as possible, jumping from roof to roof, until I've reached the Copper District.
DM: There are some steep rooftops and big gaps along the way. Make a Dex check, DC 13, and a Str check, DC 14. Failing either will result in you falling to the street below.
PC: Can I use Acrobatics and Athletics?
DM: Sure.
PC: *rolls* Passed both of them! I'm free and clear!
End Action.
Sometimes, an adjudication covering multiple instances of action is a good idea. Sometimes, the best way to do that is by calling for more than a single check.