The youtube algorithm brought this video across my feed the other day. Being as Charisma in games is something I am always interested in seeing I clicked it. I am not going to say this person is right or wrong, their advice - like all advice - should be listened to, considered, measured for usefulness in the unique situation for you, and then used or discarded. I will say however that I feel this is an incomplete argument for both how to handle Charisma, and how you can handle shy players with High Charisma characters - and high charisma players with low charisma characters.
Option 3 For Handling Charisma
In the video, Dungeon Craft presents two options for handling Charisma rolls. The first option is to RP out the conversation and then roll. The second option is to roll and then RP out the situation. Both options have problems which he lists. However, there is a third option that I find works well for me.
With the third option what you do is you ask the player "what is your approach?" after they tell you what they hope to gain with their persuasive/intimidating/deceiving attempt. If the person likes to RP they can launch into their play of their character - and I as the GM can discern their approach from that. If the person is shy, they can just tell me the generalities "I want to be this guys best buddy" or "I want to make us look and sound like the biggest threat ever."
With the player's approach in mind, I can now set a difficulty for the roll (based on how well I feel that approach would work with that NPC) and have the player roll. If they roll well, they get - or move closer - to their goal. If they roll poorly, it is not an effective strategy at this time against this NPC.
There's More to RolePlaying than Stage Acting and Speaking Verbatim
The core of my idea of RP is that there is a lot more to roleplaying than just speaking verbatim for your character. What your character does, the decisions they make, who they help, who they don't help, and basically every decision made for the character is part of the roleplay. This is how some groups have very heavy RolePlay in combat, because while their characters use effective tactics they're not necessarily the most effective tactics, but they are the tactics that character would use and does rely on.
Coaxing Out The Shy Players
Some people are shy. Some people are introverts. Some people, for whatever reason, just don't feel comfortable with the spotlight on them and the GM talking directly to them in front of the other players. And this is perfectly fine. Also, there is no reason that someone who is shy can't play a high charisma character. Just like there is no reason you can't play a character of a different gender/sexuality. Just like there is no problem for someone who is physically average or weak from playing a paragon of physical ability. Half the fun of RPGs for so many people is being able to play someone, or something, that you're not. So let them play it, and work with them to play it.
This is also where I find asking for an approach works well. The player isn't on the spot to be super charismatic, sing a song, or actually be suave/smooth. They just have to point to their favorite trope and you're good.
That Came Out Better In My Head
As for the 'high charisma' players playing low charisma, there is no reason they should benefit from their player stats unless you are giving the same benefit to other players. If you would let someone 'roleplay' past the guards without a check - or give them an advantage on the check for that roleplay - would you do the same for a player who performed an actual feat of strength in front of you? Can I come to your game, flip a couch, and have my trash tier strength mage out arm wrestle the barbarian? What if I actually beat the barbarian's player in an arm wrestling competition? No? I didn't think so. The same should be true for Charisma.
And the argument "but they presented a good case" doesn't work well either. Remember, one of the common presentations of 'low charisma' is the person thinking they actually have high charisma. So they go and try to act all smooth and suave, but come across about as fragrant as 4 month old mold in an unwashed bugbear's gym bag. We see this in real life, and in fiction all the time. The person pictures in their head - or even remembers the conversation - one way, but in actuality it came across much worse. They think they smoothly offered a bribe. In real life they leaned in, waggled their brows and went "how much coin for you to look the other way while I hit that guy with a hammer?" Or they were a bit too forward? A bit too close? Breathed hard down the person's neck? Stammered and hawed, unable to actually say what they had so gracefully picked out in their mind?
And the same is true in reverse. The same lewd joke told by someone with high charisma and low charisma will come across different. A person with low charisma walking up to someone and going "Hey, wanna smash?" will probably have the guards called on them. Someone with high charisma? Maybe they get a chuckle and a "you're a forward one" but the conversation then proceeds. Why? Because with high charisma their presentation is better, their timing is better, they have that damned smile that shows "just a bit of fun. Aren't I a charming person you want to talk to?"
That doesn't mean the high charisma character can't have a difficulty made higher by the wrong approach, but it does mean they'll get away with social faux pas that could land a less charismatic character in the dungeon when done to a king.
The Character Is Not The Player
The key to remember is the character is not the player. The character does not benefit from the player's abilities, nor should they. If that is how you run your game and have fun, that is fine. Just realize you are opening the door for socially adept players to make stronger characters as they dump stat charisma but don't suffer the full penalties of dump statting it. And if you're not giving that benefit to everyone, you're giving that player an edge over the other players too.
The choice is yours, but half the fun of a low charisma character is being awkward/bad in social situations. And when a player chooses a low charisma character, they're choosing to have that obstacle. So let them deal with it. By the same token, the person who makes a high charisma character should benefit from it.