My non-specialist take is that the linked article's explanation of the book/thesis's findings sounds confused. The thesis's abstract is pretty different from what the article seems to be saying.
The OP article is extremely confusing, and the writer itself seems confused.
My understanding is that the author of the book (Ester Salgarella) makes the following three claims:
Claim 1: Linear A and Linear B are not different scripts, but rather two different writing systems (orthographies) for the same script. She does not claim they are used to write the same language, nor that Minoan (or whatever language is written by LA) is linguistically related to Mycenaean Greek. In this sense the proposed "Aegean Linear Script" is not different than cuneiform, which has originally been developed to write the Sumerian language and has been adapted over time to write such diverse (and unrelated) languages as Akkadian (Eastern Semitic), Hittite (Indo European) and Elamite (isolate?[1]).
Claim 2: Linear A does not represent a single writing system, but in fact a collection of regional writing systems. In this context, Linear B can be viewed as just another variant of the same script, adapted to write another language. It notably comes at a later period (probably under different political conditions, i.e. under Mycenaean rule), is more standardized and does not exhibit regional variation. But we can still classify the base script itself as being the same.
These first two claims do not seem extremely controversial for me. However, there seems to be a third claim, which is implied in both the abstract of the book, but I'm not sure Salgarella is making this claim directly. That claim could prove more controversial.
Claim 3: There is a continuity between regional LA variants and the LB used by Mycenaean Greek-speaking scribes. This seems to imply that the Mycenaean scribes did make a concentrated, singular effort to borrow LA and adapt it to their own language - but rather that some regional variants of LA gradually evolved to standardize the rendering of Greek speech. I assume this started with transliterating Greek names and eventually standardizing the rules for rendering all Greek words using the Aegean Linear Script syllabograms. This system, which has gradually evolved, became standardize as what we call Linear B.
I personally think this claim makes sense if we look at the way other scripts (especially complex, logographic-syllabic scripts) have crossed between languages in the past. In the first phase you would usually have bilingual speakers writing in the script's original language and gradually developing better and more formalized adaptations for incorporating their own native language into that script. This is at the very least how Chinese characters have been adapted in Japan, where the standardization process has taken hundreds of years (in fact, you could claim that it is still not over yet).
Whether or not this claim has strong evidence I don't know, but is probably the main "breakthrough" of this research. The other ("internet-powered") breakthrough is that the SigLA database[2], I personally feel to be the coolest thing coming out of this research. Having dabbled a little bit with corporas of both deciphered and undeciphered scripts in the past, this is definitely a UX revolution. It seems like such a joy to use.
[1]: I know some theories connect Elamite to Dravidian languages through the Elamo-Dravidian family. I can't really comment on how proven that is.
