It's classic Conan, going from vagabond to king. And it happened in real life too.
But take a look at the typical Fighter, Barbarian, and Rogue: one or more of Int, Wis, and Cha are dump stats. Sure you can take the Skilled Feat, but as a king you're taking important decisions and trying to persuade important people with only that Proficiency Bonus. Sure, having your advisors to hand will give you Advantage, but when the crunch comes it's just you. You need to say the right thing at the right time. You need to make the right decisions.
So, mechanically, how do they do it? Or rather, how does your PC get to do it? I suggest that they get to add their high Proficiency Bonus a second time.
I smell a Feat here.
Prerequisite: Proficiency Bonus +4
Benefit: when making a skill roll or opposed skill check after you make the roll but before you know the result you may add your Proficiency Bonus to the roll. This ability does not stack with Expertise or any other ability which adds your Proficiency Bonus a second time. You may use this ability a number of times per day equal to your Proficiency Bonus. You regain all uses after a Long Rest.
Essentially this is a Shield spell for skill
While I think the aim is clear, I'm not happy with the phrasing. In particular it is messing with the Rogue's Expertise ability. How would you do it?
I disagree with the premise, you can run a 16 Wharisma fighter just fine and I have run a 16 Wisdom fighter. If you pick human and take the skilled feat, you start with 8 proficiencies (2 fighter, 2 background, 1 human, 3 feat). Pick a subclass that gets you another proficiency at 3rd level and now you have 9. Skill expert at 4th level get you to 10 skill proficiencies with expertise in 1 of them. Now you take a 3-level lore bard dip and you get 4 more proficiencies, expertise in 2 more of them and jack of all trades for the rest. So that is 14 total, which is the exact same number as there are combined intelligence, wisdom and charisma skills.
So at 7th level, you are proficient in every single Wisdom, Intelligence and Charisma skill, have expertise in 3 of them and you have half proficiency in all the strength and dexterity skills. You can also add fighting styles (superior technique) that can help with these checks even further.