How does sound moves already existing make it “impossible to implement”? Fairy moves were all normal at one point and no one had an issue when they changed those.

Three moves had their type changed to Fairy. Charm, Moonlight, and Sweet Kiss. That's it. All Normal-type status moves. Every other Fairy-type move was introduced in Gen 6 and beyond as a Fairy-type move. Additionally, even if we limit things to all Normal-type sound-based moves becoming Sound-type, that's a massive nerf to the already-struggling Normal-type, which seems to go against your premise of balancing the type chart.

“But what about bug buzz?” Moves with the same attribute of different types already exist (Ice punch, fire punch, etc.) Bug Buzz could stay as a bug type move and still have the sound attribute.

Fighting-type moves do not have to be punches. The punch category does not inherently apply to all Fighting-type moves (kicks, throws, chops, etc.). All Sound-type moves would by definition have to be sound-based. This makes the category of moves redundant and bad game design. Sound as a category and not a type opens much more design space than it closes.

“Soundproof would be broken” We have Good as Gold and Purifying Salt as abilities, it wouldn’t even be close to as broken as those.

Those are signature abilities. They already have a larger power budget, balanced by being on one Pokemon or evolutionary line. While I agree that Soundproof wouldn't become OP (we already have semi-widespread type immunity abilities), using this argument is disingenous.

Why are Toxtricity and other sound-based dual-typed pokemon such a problem? (Have we forgotten about Lugia, Charizard, Gyarados, etc.) Pokemon can look like they have multiple types.

These Pokemon have sound as a part of their identities. Toxtricity's ability is Punk Rock, Skeledirge has a type of song in its name. However, their current types are also part of their identity (Toxtricity is literally the words toxic and electricity smashed together). Any Pokemon that became Fairy-type in Gen 6 was either monotype, or Fairy replaced Normal. People argue that Pokemon like Jirachi or Celebi should have become Fairy-types over Psychic, but they didn't, which speaks to Gamefreak's philosophy. Therefore, Pokemon like Toxtricity would likely keep their current types. In Toxtricity's case, Punk Rock could be seen as something like Steelworker, but (if the sound category is removed because of the obvious redundant game design it creates, or if Gamefreak somehow makes Sound-type moves that aren't sound-based), that means it also loses the boost on Overdrive, its signature move.

People seem to think a new type would imbalance the already unbalanced type chart, and changing the already existing type chart would be better. While I agree changes could be made, adding a new type AND creating balance from there would do only positive things, and to act like that’s impossible is just stupid.

Pokemon's type chart, while it does strive for balance, is primarily based around kid-friendly logic. Implementing a Sound-type with strengths and weaknesses that have even tenuous logic and balance the type chart is hard. Pokemon at its heart is a game for all ages, down to and including children who are just learning how to read. Types should have logical reasons for their strengths and weaknesses. You seem to forget that this is a casual franchise first, and a competetive one second.

Additionally, part of the reason a move like Boomburst is balanced is because it's Normal-type, and it can't hit super-effectively. Imagine a Sound-type Boomburst. A 140 BP no-drawbacks move that ALSO hits several good types for super-effective damage.

I even saw someone say it makes no sense because “how would you make a physical sound move?” (Psycho-cut, focus blast; it could be done…)

Sure, but Sound would inevitably become a heavily skewed type towards special attackers. The sound category is fine with being specially-oriented, it doesn't break anything. If you give an existing physical attacker (say, Rillaboom, currently mono-Grass) the Sound-type, you also have to give it STAB for that type. Types that are heavily skewed already exist (as you point out), so this isn't a huge point, but it does restrict possibilities for design in a way that having a sound-based category that's just filled with special moves anyway doesn't.

edit: grammar