For what it's worth, I speak three languages (English, Japanese, and German) and play three instruments (trumpet, mandolin, and guitar). :)
I don't know what a "learned skill" is, because to me both programming and mathematics are learned skills. What I hear you saying is, "You can't play chess well by thinking like a programmer." That I agree with, for sure! I can't learn to program well by thinking like a mathematician, either.
That said, whether experts can express them consciously or not, there's a set of rules and heuristics by which they're making decisions. They might be incomplete, inconsistent, or highly fit to the circumstances (e.g., when I see this pattern on the board my reflex is to do X, but I don't know why), but they're still there. They might use metaphors and justifications that are foreign to a programmer, but they're still there.
For example, surely some people get better at chess more quickly than others. That is, two players with the same prior chess-playing experience both play N games. One player will probably "learn more" about chess than the other.
Why? Is it just because one player is "smarter" or has a "better intuition?" Because they're better at spotting and internalizing patterns? What does that mean? What is the faster learner doing that the slower learner isn't?
To me "intuition" is just a habit we've internalized so deeply it's hard to express in words. I'm not sure what the right level of abstraction or the right metaphors for thinking about chess are, but I imagine thinking about it in terms of territory, optionality, exposure, momentum, etc. are the most useful. I could be wrong.
My point was more that whatever a good frame of mind is, it's more effective to focus on the frame of mind and train beginners to adopt the habits of that frame of mind explicitly than it is to teach them through a million isolated examples and trust that they'll learn the most appropriate generalizations.
To take pinning, for example, we might start by talking about short-range and long-range tactics. Here's my rough draft of an explanation. Keep in mind I know virtually nothing about chess and my suggestions about what to practice might be actively developing bad habits! Hopefully you get the idea, though.
Sometimes you want to be able to influence your opponent from a distance so that your pieces are less exposed. The queen, rook, and bishop are the three pieces capable of doing this because they can move an unlimited number of spaces in at least one direction. The other pieces are limited in their movement and therefore can't exert control from a distance.
Pinning is one example of a long range tactic where one of our long-range pieces threatens a low-value piece, but the opponent can't move it without exposing a higher-value piece. For example, your bishop might be lined up to attack the opponent's queen, but one of the opponent's knights is blocking the way. In this situation your opponent can't move their knight without opening up their queen to attack. We'd say that the "bishop is pinning the knight to the queen" or "the knight is pinned to the queen by the bishop."
As a beginner it will take time to internalize all the moving parts of pinning. Remembering which pieces are worth more and recalling that without thinking takes time. Seeing patterns on the board without "manually" going through each possibility will take time. Go easy on yourself and take it slow. Don't get frustrated when you miss a chance to pin a piece. You'll develop good habits over time if you're deliberate in your approach.
To start, here's a board. Look for high value pieces and see if they're exposed in any of the directions that would be open to attack by a queen, rook, or bishop if those pieces were placed appropriately. Also look at your queen, rooks, and bishops to see where they can move. Are they able to move in a place that threatens any high-value pieces if there were no other pieces on the board?
These are your potential opportunities for pinning.
If you practice this over and over it'll become a habit and will stop feeling like "manual labor."
Take a look at this board and go through the process above. Is there some piece you can move which will pin an opponent's knight to their king? Are there any other opportunities to pin on the board? You can only make one move: if you had to choose between the various opportunities to pin, which would you choose and why? Can you see any advantages or disadvantages between the opportunities you have to pin an opponent's piece?
It's ok if you can't or if you think your reasons are no good. Give your best effort and keep in mind that your reasons might not be good reasons or the reasons an expert would give. You'll develop instincts for how to evaluate multiple possible moves as time goes on.