I have a minor improvement/variation on your proposed system. I tweaked the skills around a little bit mostly. Tools, languages, armor and weapon proficiencies map back to one of the 11 skills. Rough that you were not able to squeeze the monk into the fighter and the rogue.
Skills
This section replaces the proficiency rules, except for armor, weapons and tools. Using a weapon or tool that you are not proficient with gives you a disadvantage to the skill roll.
The three types of rolls in the game (ability check, saving throw and attack roll) are replaced by a single type: the skill roll. To make a skill roll, roll a d20, add a skill modifier and an ability modifier (STR, DEX, CON, INT, WIS, CHA) and compare the total to a target number (the AC or DC). Thus, rolling Acrobatics + DEX means rolling a d20 and adding your Acrobatics skill modifier and your Dexterity modifier.
A character starting skills modifiers are granted by his race, class and background. Below is the list of skills, the actions that use that skill modifier and the ability used for that action. For example, a rogue with +3 Subterfuge and a Intellect of 14 wants to disable a trap. He rolls a d20, adds his Subterfuge skill modifier (+3) and his Intelligence modifier (+2) for a total of d20+5 and compares it to the trap DC set by the DM. If he didn't have proficiency with Thieves' tools, he would have disadvantage on the roll.
Acrobatics (DEX): Jump, Balance, Dodge, Tightrope walk, Tumble
Athletics (STR): Run, Jump, Lift, Swim, Climb, Fly, Resist being restrained or moved
Attack (DEX): Attack, Disarm, Grapple, Touch, Trip
Craft (INT): Appraise, Craft, Repair, Engineer
Fortitude (CON): Concentrate, Resist Physical (Poison, Diseases, Polymorph, Petrification)
Influence (CHA): Bluff, Charm, Intimidate, Persuade, Perform, Lead
Lore (INT): Recall Knowledge, Research knowledge, Decipher Script, Languages
Nature (INT): Calm/Train Animals, Camouflage, Forage/Hunt, Navigate, Ride, Track, Leave no tracks
Perception (WIS): Listen/Search/Spot, Insight, Initiative, to see through illusions, ambush, sneak attacks
Subterfuge (DEX): Sneak, Sleight of hand, Escape bonds
Will (CHA): Resist Mental (Fear, Charm, Hold Person)
Tool Categories
Alchemy Kit (Craft)
Artisan Tools (Craft)
Disguise Kit (Influence)
Forgery Kit (Lore)
Gaming Kit (Influence)
Healer's Kit (Lore)
Herbalism Kit (Nature)
Musical Instrument (Influence)
Navigator's Tools (Lore or Nature)
Poisoner's Kit (Lore or Nature)
Traps and Locks Kit (Craft or Subterfuge)
Languages
Common (Lore)
Elvish (Lore)
Dwarvish (Lore)
Etc.
Weapons
Longsword (Attack)
Longbow (Attack)
Ray of Frost (Attack)
Etc.
Armor
Plate (Athletics)