My impression is that people tend to be exposed to grammar early on in school, in the form of a lot of arbitrary-seeming rules, which do not necessarily correspond with the colloquial spoken language
And indeed, they're not supposed to. "Grammar" in the sense of school consists of rules for signaling high status via speaking and writing. (The level of "arbitrariness" is what you'd expect given this.) Nothing to do with "grammar" in the sense of theoretical linguistics.
In that sense, language is value-neutral.
Linguists, however, are too hasty to jump to this conclusion in their attempt to explain that "evaluating" different language varieties is not their subject as linguists. There may be legitimate arguments (aesthetic, utilitarian, etc.) for why some forms of language are "better" than others; it's just that such arguments are strictly irrelevant from the point of view of theoretical linguistics (though not sociolingustics, etc).
rules for signaling high status via speaking and writing.
And clarity.
Nothing to do with "grammar" in the sense of theoretical linguistics.
In English, Irish and French classes I learned about parts of speech, regular and irregular verbs, cases and declensions, moods and tenses and conjugations, gender and agreement etc etc. These are not really theoretical linguistics, but they are prerequisites for it.
If you've spent any time with foreigners learning your language, you may have been in conversations like this:
People can't automatically state the rules underlying language, even though they follow them perfectly in their daily speech. I've been made especially aware of this when teaching French to Chinese students, where I had to frequently revise my explanation, or just say "sorry, I don't know what the rule is for this case, you'll just have to memorize it". You learn separately how to speak the language and how to apply the rules.
Morality is similar: we feel what's wrong and what's right, but may not be able to formulate the underlying rules. And when we do, we're likely to get it wrong the first time. For example you might say:
But unlike grammar, people don't always agree on right and wrong : if Alfred unintentionally harms Barry, Barry is more likely to think that what Alfred did was morally wrong, even if both started off with similar moral intuitions. So if you come up with an explanation and insist it's the definition of morality, you can't be "proven wrong" nearly as easily as on grammar. You may even insist your explanation is true, and adjust your behavior accordingly, as some religious fanatics seem to do ("what is moral is what God said" being a quite common rule people come up with to explain morality).
So: beware of your own explanations. Morality is a complex topic, you're even more likely to shoot yourself in the foot than with grammar, and even less likely to realize that you're wrong.
(edit) Related posts by Eliezer: Fake Justification, Fake Selfishness, Fake Morality.