Okay, but what does Insight do?
Fine. That's better.
First, just to make certain there is no misunderstanding, when I say "Passive" I mean "always on, in the background". If you're interacting with someone who lies to you, you always have a BS-detector going. Something perhaps just doesn't jive or feel right, it is your instinctive "gut feeling." Empathy and understanding how someone else is feeling or understanding what their goals might be is also passive and involves Insight.
In the same respect, Perception is your senses. They are also passive ("always on"). If I am having a conversation and someone nearby opens the door, I will probably still hear it, just as if someone steps on a squeaky floor board, etc.
By making these passive, I can have them contested by things like Influence (Deception or whatever) and Stealth (to move quietly). If the rolls fail to beat the passive scores, the PC senses something is going on. In the case of Insight, perhaps the person trying to influence you is lying, or telling the truth but has ulterior motives, or whatever. In the case of Stealth, the "sneaker" is possibly heard or seen depending on the environment and situation.
If your passive scores beat the roll, you can use Investigation to learn more. Now, you are trying to read the tells of the person. Are they simply lying? Holding back information? Have a different agenda? Why are they trying to influence you? Against Stealth, do you see the sneaker or just a form and know something is there? Which direction did the sound come from? How far away was it? In short, Investigation is used to gain more information than your passive scores reveal, and it is an active attempt to discern that information.
For example, suppose two guards are on watch with WIS 10 and 12. You make a Dexterity (Stealth) check to sneak past them in the nearby shadows. You roll a total of 11, beating one guard by not the other. You overhear one guard, "Hey, did you hear that?" "No," replies the other.
But at this point, I as the DM, can have both guards roll Intelligence (Investigation) as they remain quiet, listen intently, and scan the shadows to determine if they see anything out of the ordinary. I roll for both and get 12 and 15, both beating your stealth. Now, they either heard you or saw you (perhaps something metal glinted off of the moonlight) and you see them moving towards you. What will you do???
If they had failed on their investigation rolls, you would have remained quite enough and/or hidden enough they didn't notice you, and so would resume their guarding and you could move on successfully.
This assumes insight will always be assigned to wisdom, though, which I think is a mistake. Skills should float between attributes more often rather than less.
That's fine, and I float them all the time if it fits the narrative. However,
Wisdom reflects how attuned you are to the world around you and represents perceptiveness and intuition. IMO Insight and intuition are synonymous really.
If you want to understand something or figure it out, then yes it is Intelligence IMO but also falls under the Investigation skill as
you look around for clues and make deductions based on those clues. This includes looking for indications someone is lying to you or hiding something from you. Wisdom tells you something is up. Intelligence lets you figure out exactly what is up.
That is why I do it this way.
and the odd weakening of STR-based PCs who are already generally the weakest on the skill front (yet I weirdly like the way you divided up the STR skills).
We did this because Athletics is too good and covers too much. You still have Strength to base this on... so if you have a STR 18 or 20 you will be "good" at all these things already IMO. Skill proficiency represents additional training and dedication. Plenty of people are great at climbing or lifting or swimming or wrestling (grapple) or running or whatever, and they tend to be good at many other things if they are great at one or two. BUT very few people are ever great at
all such Strength/Athletics activities.
For example, as a Rogue I often take Athletics (in RAW games), but what am I
really taking it for: climbing and running, maybe swimming. I am not taking it to be better at breaking down doors or grappling someone typically. This means my Rogue with STR 12 has +3 on Athletics checks, but your Fighter with STR 18 is still better at +4. Assuming you bump STR at 4th or 6th levels, the Rogue will be +4, but the Fighter is +5. Often, it'll be 9th level or higher before I can even
match your bonus. Now, of course I could take Expertise in Athletics, but then we are talking a whole other level of dedication. At any rate, athletics, itself, is still broad enough as a catch-all, but should not include training in brawn or grapple tasks IMO.
This also forces martial types to be more specialized or give up another skill if they really want to be Strength-oriented.