What is a believable in-game explanation of GUMSHOE’s spending point system?

I am studying the GUMSHOE system as it is implemented in Trail of Cthulhu and, while I can grasp the logic behind its mechanics, I have a hard time of translating the rules of allocating points to skills and expending them into a believable in-game analogy, in order to minimize their meta-gaming aspect.

As far as the d100 (BRP) system is concerned, a 40% chance of succeeding in e.g. Archaeology could be translated as the chance of the player possessing a relevant knowledge. There is a believable in-game analogy: the higher the percentage, the higher your expertise and the better the chances of recalling/knowing things related to your skill. And this analogy can be consistently used in every situation.

In GUMSHOE’s terms, when a player is an expert in Archaeology, they automatically succeed in finding a relevant clue if they associate their investigative skill with the required action. In this case, there is also a believable in-game explanation; the investigators are experts, they are supposed to spot relevant clues. The immersion does not break.

But when it comes to the spending point system, they only in-game analogy I can find is that expending a point from an investigative skill could mean that players put an extra effort in order to spot additional clues, like using your Archaeology skill to estimate the origin of an ancient statue and spending one point in order to associate it with a specific famous ancient figure. But in that case, from an in-game perspective, having no other points to spend means that for some unexplainable reason you are no longer able to make extra efforts of extracting further clues when using your investigative skill.

Another approach could be that these points represent the "stamina" of your character, so expending them would mean that they become more and more "fatiqued", either mentally or physically, and thus no longer able to perform harder tasks unless they replenish them. Which is also highly unconvincing.

Since I am mainly interested in Trail of Cthulhu, I thought of providing a more mystical interpretation, like that these points represent the will of the players to overcome the obstacles lied to them by unknown forces. So unknowingly to them, there is an subconscious, telepathic "battle of wits" taking place and the more they spend their points the lesser their will power becomes. Which is also something that the characters are not supposed to be aware of; thus we still have to resort to meta-gaming.

Perhaps I am overthinking this and I can just appreciate the novelty that the GUMSHOE system brings to the investigation-driven rpgs without trying to shoehorn every rule into an in-game perspective. But even if I ignore the meta-gaming aspect, the whole point system seems very unconnected to any physical or mental characteristic of the characters, albeit undeniably effective in building tension and giving the feeling that the stakes are being steadily raised.

To summarize; what would be a plausible in-game representation of the spending point system, at least as far as the investigative skills are concerned?