A few facts, like how often Fighters really get ASI's, and the sound quality (it's like the mike is being tapped every few seconds) can be quibbled about, but overall, I liked the video.
The problem being seen here is that the game doesn't really explain "what is roleplaying". Simply put, it's taking on the role of a character, and acting as if you were them.
If you're a brave Dwarven warrior, you may act tough, gruff, boisterous, or battle wise. You may have stories from long years serving in the armed forces, or as a mercenary. You could have practical experience with all sorts of weapons, armor, and tactics.
Right up until you're expected to "prove" this with a die roll. Then suddenly, all the details you've used to bring your character to life are brought into question. Do you have proficiency in skill X? Is your ability modifier low?
These are certainly problems, but the biggest factor you're up against is the mighty d20. Even a wizard with proficiency and an Intelligence of 19 can roll low, and a total dunce can roll high- this great variance means that high bonuses can still fail you.
What most DM's need to learn is the art of not rolling dice. Is it necessary to ask the Fighter if he knows something about actual war, battle, or strategy? Probably not, and yet, he's often asked to do so anyways.
Because, and this is the kicker, really, most classes don't have much to work with when they are asked to roll dice. You're going to have a low stat somewhere. You can't be proficient in all skills. Sure, you might have expertise in a couple skills, or some extra proficiencies, but each character is good at what they are good at by design. And if your Rogue has no real use for Charisma, then he isn't going to be any more dashing or suave than the typical Fighter.
An argument can be made that social skills actually subvert roleplaying- you can give an impassioned speech in character, but if the DM decides to make you roll, the success or failure is now up to a die, not your acting ability.
Sure, a good DM might give you a bonus- but you can always roll a 2. But the counter argument is, if the DM doesn't ask for rolls because you're a good roleplayer, then your actual abilities no longer matter, allowing you to succeed more often than players who might have better numbers, but better acting ability.
I mean, as an aside, most die rolls, you roll and then you see if you succeed or fail and roleplay accordingly. But have you ever noticed most social die rolls work in reverse? You are asked to roleplay first, then see if your effort mattered!
So I don't think it's the Fighter that has the problem here (though they could use a little more oomph), it's the fact that the game itself is fairly shabby when it comes to how it handles social encounters.
You can either try to convince the DM, or you can try to roll a die and hit some arbitrary (likely high) DC. Neither approach seems ideal. And worse, you might roll a 27 and still find an NPC resistant to your attempts at persuasion!
"He's a warlord who has conquered nations, he's not going to be afraid of a level 3 guy who can't even afford plate mail!"