> aren't there a couple of core features/structures all languages have?
The answer is 'no', especially in the sense that there's a relatively finite set of possible grammars that a putative natural language could be compared against. In terms of basic parts of speech, I believe that every language does have something that you can describe as a noun, but that's more or less the only "universal" part of speech (there are some languages that essentially don't have verbs--you "do a look" rather than "see", e.g.).
The more serious problem, I think, is that the corpus of Linear A is simply too tiny to do any serious study, and I don't know how well the written corpus is at actually reflecting problems like segmenting text into words or morphemes. In essence, the available evidence is so paltry that you could justify just about any grammatical hypothesis, I suspect.
If I'm understanding Chomskyian linguistics correctly (that's a really big if), there was originally thought to be an inherent "language organ" that strongly controlled grammar. But over time, and as linguistics documented more languages, the things that are universal in grammars in this subdiscipline has essentially been reduced to 'merge', which is an abstract concept that I'm pretty sure I don't understand.