Yes, Latin words CAN go anywhere. But practically, they didn't, except for emphasis (and in poetry).
There's an advantage to learning the way Romans actually constructed sentences because it helps understand shades of meaning. Also, if not, you'd be speaking/writing an odd, Yoda-like language, that could technically be understood but is wholly unnatural.
I guess the word order and grammar varied quite a bit depending on which extremity of the empire the language was spoken in. For example, I presume the northern dialects that ended up becoming the langues d'oïl (like French) didn't usually drop the pronouns, whereas the southern dialects were more lax in that regard. In its written form, all eyes were probably on Rome. I definitely agree with you, though.