>Agglutinative languages, like Finnish or Hungarian, where word order is largely irrelevant and conjugations, not word order, determine meaning.
I'd argue that word order is just as relevant (and indeed constrained by meaning) in Hungarian or Finnish as it is in, say, English, though the details of how each language's syntax works are very different. Also, there are plenty of languages not usually characterized as "agglutinative"(e.g. Russian) in which affixes rather than word order are the primary determinant of the role that a word plays.
"Agglutinative" is really just a way of saying that meaning-bearing morphemes in the language tend to be more closely bound together: where in English, for example, we'd express "in our houses" with 3 words, whereas in Hungarian, these would form a single word "házainkban" (ház[a] "house", -i- plural, -nk "our," -ban "in"). We know they're separate words in English because other stuff can come between them ("in each of our big houses"); not so in Hungarian. (So arguably agglutinativity is more about constrained morpheme order than it is about free word order!)
>Topic/subject languages, like Japanese, where previously defined context can radically alter meaning. "Watashi wa hamburger desu" can mean "I'm a hamburger", "I'll have a hamburger", "If you ask me, it's a hamburger", etc.
Of course context can radically alter meaning in English too; think of the waiter who says "which one of you is the hamburger?" ("Me, I'm the hamburger, and he's the Caesar salad.")
Almost all language classifications exist along a spectrum, or perhaps it'd be better to talk about a high-dimensional space. Different languages rely more or less on different strategies, but there are almost no linguistic features for which analogues can't be found in most languages.
I absolutely agree that people should be able to choose how they use pronouns (or in fact any component of language). Communication is a mutual endeavour, and I will always make my best good-faith effort to understand what an interlocutor is trying to convey (assuming I want to talk to them in the first place :), even if they choose to use language differently from the way I would. In fact (within reason) I will even adapt my own usage in some cases - e.g. I will do my best to use the pronoun "they" (or "xe" or "foobar" or whatever) when referring to someone who wishes to be referred to in that way, even though (despite knowing several languages with gender-neutral pronouns) I find that takes non-insignificant mental effort when I speak English.
That said, precisely because communication is a mutual endeavour, and language is a coöperative framework, there is no guarantee that -- just because you choose to use "they" in this way -- others will follow. I wouldn't be terribly surprised if, within a generation, "they" became an unmarked definite gender-neutral pronoun in English (today it is arguably unmarked in the case of an indefinite antecedent, but extremely marked otherwise). But I wouldn't be terribly surprised if it didn't, either. Language change is an emergent process and not something that can easily be forced in either conservative or radical directions.
The problem with "they" is that it is in fact a very bad translation for "ő" in a sentence like "ő mosogat." We typically find gender-neutral "they" in English when there is an indefinite antecedent like "someone" or "anyone" - "if anyone objects, they should talk to me." In contrast, in Hungarian, including the third-person pronoun "ő" means the speaker has a very definite subject in mind and in fact wishes to emphasize that (a neutral "he/she is washing up" would be just "mosogat", without any explicit pronoun; "ő mosogat" almost means "HE [not someone else] is washing up." (Or "SHE" of course.) The only case where I can imagine translating "ő" as "they" in this kind of context is when I know that the person in question explicitly prefers the pronoun "they," e.g. because they identify as non-binary.
I was amused to find that "ő orvos. ő sebész. ő agysebész. ő kardiológus. ő pszichiater. ő radiológus. ő bőrgyógyász. ő nőgyógyász. ő szülész. ő fogorvos." was translated as "she is a doctor. she is a surgeon. he is a brain surgeon. she is a cardiologist. she is a psychiatrist. he is a radiologist. she is a dermatologist. she is a gynecologist. she is a midwife. she is a dentist." The (small) majority of physicians in Hungary are in fact female -- I don't know how that breaks down by specialty though. If you are to believe Google Translate, only brain surgeons and radiologists are male by default! (By the way, "szülész" to me means "obstetrician," not "midwife.")
I'd argue that word order is just as relevant (and indeed constrained by meaning) in Hungarian or Finnish as it is in, say, English, though the details of how each language's syntax works are very different. Also, there are plenty of languages not usually characterized as "agglutinative"(e.g. Russian) in which affixes rather than word order are the primary determinant of the role that a word plays.
"Agglutinative" is really just a way of saying that meaning-bearing morphemes in the language tend to be more closely bound together: where in English, for example, we'd express "in our houses" with 3 words, whereas in Hungarian, these would form a single word "házainkban" (ház[a] "house", -i- plural, -nk "our," -ban "in"). We know they're separate words in English because other stuff can come between them ("in each of our big houses"); not so in Hungarian. (So arguably agglutinativity is more about constrained morpheme order than it is about free word order!)
>Topic/subject languages, like Japanese, where previously defined context can radically alter meaning. "Watashi wa hamburger desu" can mean "I'm a hamburger", "I'll have a hamburger", "If you ask me, it's a hamburger", etc.
Of course context can radically alter meaning in English too; think of the waiter who says "which one of you is the hamburger?" ("Me, I'm the hamburger, and he's the Caesar salad.")
Almost all language classifications exist along a spectrum, or perhaps it'd be better to talk about a high-dimensional space. Different languages rely more or less on different strategies, but there are almost no linguistic features for which analogues can't be found in most languages.