But what about languages like Latin, where a complicated system of inflection allows for much greater if not total syntactic freedom? Considering that most other European languages have simultaneously moved toward rigid syntax rules while most of the inflections have disappeared, how do those languages fit into the above scenario? Is it possible that some precursor of PIE had simple morphology and rigid syntax, like English, and then acquired the seven or so declensions, four or five cases, and three grammatical genders characterize, say, Lithuanian?
I think you’re conflating syntax and word order here. Latin speakers were not constrained by a rigid word order, but that does not mean you can say that Latin is characterized by greater syntactic freedom than other languages.
I’ve read that the Insular Celtic languages (but not Continental Celtic) have VSO word order. Since changes in word order are rarish and often arise via contact, it is hypothesized that languages of the British Isles prior to the Celtics may have had VSO word order.
Since some Semitic and related North African languages (Berber, etc.) are the only “nearby” languages with VSO order, this leads to speculation about the prehistory of Western Europe. (With only a smallish number of plausible Afroasiatic–>Celtic/Germanic cognates, I think such speculations are considered fringish.)