I agree; GP didn’t choose the best examples for this. But there are languages like Yimas, Tiwi and Dyirbal for which word order truly is insignificant, so the point still stands.
There's a difference between "free ordering of constituent phrases", where you may rearrange subject noun phrases, object noun phrases, adverbial phrases and the verb into any order depending on emphasis etc., and completely free word order - like in the languages you mentioned - where "the little man walked through the big door" could end up as "man door through walked big little the the" or something like that, completely breaking apart what we'd consider "constituent phrases" in a language like English, where "the little man" etc. always move together in a sentence.
> There's a difference between "free ordering of constituent phrases", where you may rearrange subject noun phrases, object noun phrases, adverbial phrases and the verb into any order depending on emphasis etc., and completely free word order - like in the languages you mentioned - where "the little man walked through the big door" could end up as "man door through walked big little the the" or something like that, completely breaking apart what we'd consider "constituent phrases" in a language like English, where "the little man" etc. always move together in a sentence.
But how common is it to have the former without the latter? Most languages I read about have either both or neither.
Many language seem to have at least a somewhat flexible ordering of constituents. It may often depend on emphasis, information structure, etc. German for example can be quite flexible in that regard, and the effects this has on information structure are rather subtle (however, the verb always has to go in the second position, so it's not totally free). There are probably languages where it's even more free, but I don't have any examples off the top of my head. By contrast, there are languages where word order is very rigid (even in English, while you can rearrange constituents up to a point, it's not nearly as common as in German, and I would argue that it can often sound slightly archaic)
This I don’t agree with.† The rest of your post, however, I agree with entirely, and how I forgot the examples you mention I’m not entirely sure.
[† see what I did there?]
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.