For my two current CoS campaigns I also futzed with skills quite a bit, so I definitely know where you are coming from. Here's a couple things I found when I set my stuff up:
- Background was the second thing players chose during character creation, before choosing their Class. The order went Race -> Background -> Class. Not only did it make sense from a progression standpoint (choosing what they were when they were born, to what they did before adventuring, to what they are now that they are an adventurer)... but it also eliminated the "doubling up" of skills. They got their Background skills first, then when they chose their Class, they chose other skills from the class list that they didn't yet have. Thus no doubling up ever resulted and no one got to chose "anything" that they wanted.
- I use the Variant Rule in the DMG where I do not align skills to abilities. I ask for an ability check, and then if a skill might apply, they can add their prof bonus to it. As a result of this, I didn't need to have skills cross over onto two lists (or however many), because I don't actually have set lists. Which thus eliminates what you've needed to do with Intimidate or Gather Information for example.
- Due to this non-assignment of skill to ability score, it impacted which skills I used, added, and eliminated. For instance, Intimidation was removed as a skill, and if someone tries to threaten someone (like by grabbing them around the throat), I'll assign it as a STR (Persuasion) check. By the same token, I removed Acrobatics, and instead any balancing or tumbling is now a DEX (Athletics) check. My skill list includes:
Animal Handling
Arcana
Athletics (includes Acrobatics, now a DEX (Athletics) check)
Commerce (any actions related to money, finance, appraisal or bartering)
Deception (includes Sleight of Hand, now a DEX (Deception) check)
Etiquette (anything related to the nobility or high society, both in action and knowledge)
Folklore (anything related to the common folk or low society, both in action and knowledge)
History
Insight
Investigation (basically what you use Dungeoneering for-- finding inanimate hidden things like traps or secret doors)
Nature
Perception (finding hidden people / creatures / living things that have taken the Hide action)
Persuasion (includes Intimidation, now a STR (Persuasion) check; includes Perform as well)
Religion
Stealth
Survival (includes Medicine, all humanoid medicinal or body knowledge or action falls under Survival)
- I retained Tools as something to also gain proficiency in, mainly because I use expanded Crafting rules. So to perfom Alchemy, Herbalism, Tinkering, Poisoning, and Smithing (to gain the resultant crafted works) you make applicable ability checks plus your prof bonus if you are proficient in the Tools. And I removed the Tool proficiencies that don't give you anything and instead can just fall under other skills (so Vehicles gone, Navigation Tools gone, Disguise Kit gone). I did retain Thieves Tools because while I could have just reincorporated Thievery as a skill like you did... I figured since I was retaining other Tools for crafting, then keeping Thieves Tools didn't cause any issues.
All in all, I've been happy with a lot of what I've done for these campaigns and they seem to work well. I don't expect you'll have much issue with your changes as well, since Skills are one of the easiest things to change around and one of the most obvious parts of the game to know when you don't like how the standard version works. Best of luck!
- Background was the second thing players chose during character creation, before choosing their Class. The order went Race -> Background -> Class. Not only did it make sense from a progression standpoint (choosing what they were when they were born, to what they did before adventuring, to what they are now that they are an adventurer)... but it also eliminated the "doubling up" of skills. They got their Background skills first, then when they chose their Class, they chose other skills from the class list that they didn't yet have. Thus no doubling up ever resulted and no one got to chose "anything" that they wanted.
- I use the Variant Rule in the DMG where I do not align skills to abilities. I ask for an ability check, and then if a skill might apply, they can add their prof bonus to it. As a result of this, I didn't need to have skills cross over onto two lists (or however many), because I don't actually have set lists. Which thus eliminates what you've needed to do with Intimidate or Gather Information for example.
- Due to this non-assignment of skill to ability score, it impacted which skills I used, added, and eliminated. For instance, Intimidation was removed as a skill, and if someone tries to threaten someone (like by grabbing them around the throat), I'll assign it as a STR (Persuasion) check. By the same token, I removed Acrobatics, and instead any balancing or tumbling is now a DEX (Athletics) check. My skill list includes:
Animal Handling
Arcana
Athletics (includes Acrobatics, now a DEX (Athletics) check)
Commerce (any actions related to money, finance, appraisal or bartering)
Deception (includes Sleight of Hand, now a DEX (Deception) check)
Etiquette (anything related to the nobility or high society, both in action and knowledge)
Folklore (anything related to the common folk or low society, both in action and knowledge)
History
Insight
Investigation (basically what you use Dungeoneering for-- finding inanimate hidden things like traps or secret doors)
Nature
Perception (finding hidden people / creatures / living things that have taken the Hide action)
Persuasion (includes Intimidation, now a STR (Persuasion) check; includes Perform as well)
Religion
Stealth
Survival (includes Medicine, all humanoid medicinal or body knowledge or action falls under Survival)
- I retained Tools as something to also gain proficiency in, mainly because I use expanded Crafting rules. So to perfom Alchemy, Herbalism, Tinkering, Poisoning, and Smithing (to gain the resultant crafted works) you make applicable ability checks plus your prof bonus if you are proficient in the Tools. And I removed the Tool proficiencies that don't give you anything and instead can just fall under other skills (so Vehicles gone, Navigation Tools gone, Disguise Kit gone). I did retain Thieves Tools because while I could have just reincorporated Thievery as a skill like you did... I figured since I was retaining other Tools for crafting, then keeping Thieves Tools didn't cause any issues.
All in all, I've been happy with a lot of what I've done for these campaigns and they seem to work well. I don't expect you'll have much issue with your changes as well, since Skills are one of the easiest things to change around and one of the most obvious parts of the game to know when you don't like how the standard version works. Best of luck!