Prescriptivists you mean. Descriptivists describe language phenomena as they are without passing judgment on their validity whereas prescriptivists have certain assumptions on what constitutes parts of the language to be valid and prescribe meaning to them

"Cromulent" was a perfectly cromulent word as soon as someone thought of it.

While, at the same time, don't be mad at people that don't understand the word you used because they lack the context. Be educational, don't gatekeep.

Several languages have this. Spanish has the Real Academia Española (RAE) and French has something similar.

But they're not generally in much of a different position than a dictionary is. If the people start using the language in new ways they have little recourse other than to accept it and amend their rules. If they refuse they'll look antiquated and people start to question their influence.

They certainly do have influence of course, but the ultimate authority is the people who speak the language in the end.

Do they monitor your private messages and fine you for typos or do they just codify the language which is taught in schools and used by the authorities? If it's anything like German language regulation then it's the latter and the way people actually talk and write slowly is adapted by the language regulations.

Literal grammar police? What are the consequences for breaking word laws?

I would argue the main benefits are to teach people how to effectively switch registers as the context demands, and to expose them to a range of language they likely wouldn't ordinarily encounter in their daily lives. English teachers could do to lose the judgmental aspect of "This is the one true way to speak English, the way you talk amongst yourselves is wrong and you need do stop," but there's a definite value in teaching students, "This is a way to write/speak clearly and effectively that will be understood by quite nearly every other educated English speaker you might encounter."

As far as exposure to a broader range of language than one normally encounters in their life, I saw the importance of this first hand with many of my coworkers who were heritage speakers of Spanish. It's not my native language, but it was my primary work language for a good 5 years, and I wound up getting put on interpretation duties for our safety meetings over a native speaker with pretty limited formal education in Spanish. For topics to do with daily life, family, friends, etc, this guy would be able to speak much more naturally than I could. I might not say something that was exactly wrong, but perhaps I would be too formal or make odd word choices he wouldn't. The problem was, he completely lacked any technical and professional vocabulary, and had no concept of what words/phrases were unique to his own country and what alternatives might be more widely understood.

We would have safety meetings once a night, and they would have topics like, "When a forklift has its forks in the air, don't walk beneath it, as hydraulic failure could lead to injury or death.". He translated that one night as "Cuando la vaina del pasillo tiene esa vaina de en frente en el aire, no pasen por debajo de la vaina. Es peligroso." Basically "When the thing in the hall has the thing in front in the air, don't walk under the thing. It's dangerous." Best case, he might say "El forlift," but he would never land on "el montacargas," or even think to look it up. Some of his wilder attempts at interpretation didn't work for anyone, and the ones where he just used a Spanglish version of technical terms only worked for other coworkers who already knew at least a bit of English, and probably didn't really need the translation that much to begin with. Unfortunately, we had a fair number of employees who were monolingual Spanish speakers that he found himself just completely unable to communicate with effectively.

Granted, not everyone takes full advantage of it, but English classes do (or at least should) expose you to a broad range of the language, as it's used in various contexts and forms, while also furnishing students with the ability to expand upon that and adapt to new contexts on their own in the future. Failure to do so leaves students with stunted linguistic and communicative abilities.