I think that explanation proves perfectly that it can be translated.
Depending on the context, you'd want either explanation. If you're learning Japanese you'd probably want the explanation of how and when it's used, and the literal words that compose it.
But if you were translating a manga book and it used that phrase, you would probably just put "Ahh, we're finally here!"
I think there's just something of a bastardized use of the word "untranslatable". As long as a language can convey abstract concepts, you should be able to translate anything into it.
I think when people say "untranslatable", there's an implied "without breaking the flow or losing nuance" there. Of course you can in principle translate anything into any language, it's sometimes impossible to translate something without either a lengthy digression, or giving up on translating the full nuance. Poetry is notoriously hard to translate because of stuff like that.
I don't know that I would say this is a bastardization of the word, but I see where you're coming from.
I think we actually agree here: it's not "untranslatable", it just does not translate easily or well.
But regarding translating it as "we're finally here", that's not what it means: it's not an expression of satisfaction, it's acknowledging that to others that it's been a long trip for them.
Depending on the context, you'd want either explanation. If you're learning Japanese you'd probably want the explanation of how and when it's used, and the literal words that compose it.
But if you were translating a manga book and it used that phrase, you would probably just put "Ahh, we're finally here!"
I think there's just something of a bastardized use of the word "untranslatable". As long as a language can convey abstract concepts, you should be able to translate anything into it.