"What remains uncontested is the product of his meditations, the Samkhya Sutras, which laid the foundation for the earliest “complete” system of philosophy derived from The Vedas. The system’s central axioms would be developed over the course of a millennium..."
Is wrong for several reasons. As a Samkhyavadin of the arithmetic kind, let me count some of them:
1. What's 'uncontested' would be strongly contested by both their classical interlocutors such as the Mimansakas and by modern scholars - for example, for linguistic reasons mentioned in one of the other comments.
2. Not sure what 'complete' philososophy means, but we might assume that some account of reason and logic is part of a complete philosophical system, in which case the Samkhyavadins like many of their counterparts, took the lead from the Naiyayikas, who were closely related but distinct. Same for grammar, where everyone took their lead from Panini and the grammarians.
3. I think calling it 'Vedic Philosophy' doesn't do justice either to the Vedas or to Philosophy. For example, the accurate, elaborate and intricate performance of ritual action is central to the Vedic experience. We can't reduce that to philosophical beliefs about dualism or monism without serious harm to the original practices. It's a sign of modernity that beliefs (such as dualism) are given precedence over ritual performance.
3. Words like 'derive' and 'axioms' suggest an overly mathematical relationship which would be impossible to justify. Even the Upanisads aren't derived from the Vedas in any axiomatic sense. The Prakriti-Purusa dualism finds a precedent in the Rig Veda which says "Two birds associated together, and mutual friends, take refuge in the same tree; one of them eats the sweet fig; the other abstaining from food, merely looks on" - try deriving the Soul from that imagery.
In fact, the most well known Samkhya (influenced) text isn't the Karika but the Mahabharata, including the Gita, with Arjuna playing the role of Prakriti and Krishna that of Purusa. Which is why - to use a deductive argument - Arjuna fights even when he doesn't want to and Krishna doesn't even when wants to.
I know I am being pedantic, but I find that these kinds of reductive Whig histories of Indian knowledge traditions perpetuate the problems they are trying to remove. Far more interesting, say, from the perspective of modern philosophy of mind, is the somewhat technical question: why did most Indian traditions consider mind, aka Manas, to be a physical entity? What does it say about their account of knowledge since Manas is an Indriya, i.e., an instrument of knowledge. What does it mean for knowledge to be physical and yet normative, i.e., how can something physical be true or false?"
To reduce these subtleties to 'Hindu Philosophy' or even to six schools of Astika philosophy is deeply problematic.