Other of ideas caught my attention.
Always applies to every campaign:
Acrobatics is removed. Use Dexterity (Athletics) instead.
Sleight of Hand is removed. Use Dexterity (Deception) instead.
Animal Handling is removed. Use Wisdom (Nature) instead.
Intimidation is removed. Use Strength (Persuasion) instead.
Medicine is removed. Use Survival for any necessary checks instead.
Performance is removed. Use Persuasion instead.
Perception applies only to finding living creatures that are hiding.
Investigation applies to finding hidden inanimate objects-- secret doors, traps, hidden objects.
Mechanics added to skill list. Skill used for engineering information and as a replacement for Thieve's Tools.
Then for specific campaigns I have included various combinations of the following skills:
Warfare. Any actions, information and lore related to military action.
Nautics. Any actions, information and lore related to naval and ship action and the sea.
Dungeoneering. Any actions, information and lore related to the underground, Underdark, and dungeon crawling.
Commerce. Any actions, information and lore related to appraisal, trade, haggling, and all financial work.
Politics. Any actions, information and lore related to law and the political arena.
Definitely delete Acrobatics!
You mention interesting uses of skills. I am comparing notes.
I get much use out of the Sleight skill. The word 'sleight' (not to be confused with 'slight'), literally means 'slyness', but only refers to extraordinary use of manual dexterity. I use Sleight to pick locks, and being proficient in both Sleight and Thieves toolset grants advantage to checks. But Sleight is also for any kind of manual dexterity check, including piloting a spaceship (as that sometimes does happen in D&D). Sometimes for (Celtic) flavor reasons, I require alchemy or ceremonial magic to use ingredients extremely precisely, and require a Dexterity (Sleight) check to determine success. Anything hands can do, Sleight can do better.
Animal Handling. I started using it for 'zoology' checks. But just call the skill 'Animals' so the name can fit, heh, when writing it in the character sheet. I am also wondering if the 'handling' part should be a Charisma (Animals) check, in the sense of social skills with animals.
Maybe it is better to remove Intimidation. Currently, I am trying to make it work by making it responsible for keeping morale in ones group, when facing danger. Essentially, Intimidate is the 'courage' and 'confidence' skill. I am still trying to finetune the morale mechanics, generally a Wisdom save when a team member gets bloodied or downed. It occurred to me, in the sense of saves versus Fear and Insanity, give Intimidation, morale, and fearlessness in general to Wisdom.
I normally give Insight to Charisma as part of people skills. But sanity requires self-insight, which still makes sense as a Wisdom (Insight) check to discern if oneself is being sane. Insight is my go-to skill for psychology and psionics, normally Charisma, but other abilities are situationally possible.
In gameplay, I tend to use Charisma (History) for 'politics' and 'politicking'. History is my go-to for social sciences. For example, I use History to decipher a language. I use Insight for personal psychology. Together these two skills are the human sciences.
Commerce. I have been using the skills relating to the specific items and services, for marketplace stuff. But yeah, commerce might deserve its own skill, especially when trade and markets can be central themes in some urban settings.
I use Medicine for anatomy and all kinds biological science, especially for Humanoids. I also use Medicine for skill checks to create and use poison effectively.
I just recently deleted Perception. I make Intelligence handle ALL perception checks. The skill depends on what the character is trying to perceive. If trying to identify the scent of a potion, the perception check is Intelligence (Arcana). If trying to find a secret door, the perception check was Intelligence (Nature), but will now be Intelligence (Wrightcraft). To find a hidden person, I use Intelligence (Stealth). People who know how to be stealthy are also better trained to know what to look for when others are being stealthy.
Investigation has evolved into my check for all things 'reasoning', 'intuiting', 'piecing clues together', 'researching in a library', figuring out by trial and error an alien technology. Sometimes Investigation is the 'DM gives a hint' check. I dont use it for any particular subject matter. It is more like an Athletics check of the mindscape.
Mechanics, yeah. That is a skill that feels missing from the official skills list. I was using Nature for it in the sense the physical sciences. But I am using Nature for too many unrelated things, including ecology, elemental alchemy, Feywild, Ethereal Plane, and animism generally. For the Mechanics skill, I plan on using the name 'Wrightcraft'. Simple lockpicking is more about precise hand movements. But any kind of complex 'mechanical apperatus' would be Mechanics. Maybe even Dexterity (Mechanics), albeit Mechanics typically employs Intelligence.
Warfare skill is a good idea. I have been using History, but military history probably merits its own skill and has combat implications.
Nautics. I have been using Survival for nautics and dungeoneering, in the sense of wilderness and navigation. But I am reconsidering it.
I want to mention, I treat mounts (!) as if a toolset. So a person who is expert in all things 'Horse', has this species as a mount proficiency (= tool proficiency). A person of the elven High culture might have 'Griffon' as a mount proficiency. Similarly for vehicles. Some Norsesque characters have a longship (including similar boats) as a vehicle proficiency (= tool proficiency).