If the Universe did start from a single point, then wouldn't all particles be fundamentally entangled? How then could there be a truly "pure" state?
Pure states are a convenient abstraction for studying tiny, specially-prepared quantum systems. States of complicated systems are never pure.
One makes (very small) systems pure by careful preparation. For example, a simple spin by means of a Stern-Gerlach magnet and a screen where only the up particles can pass. More complex systems need more complex preparation to make them (at least approximately) pure.
Note also that pure states can be entangled; indeed, entanglement is usually defined only for pure states. Thus there is no conflict between entanglement and pureness.