Here's my attempt:
Huw Price has said of Reichenbach in general that he was talking about, and made clear(ish), that, the local thermodynamic gradient does not give a global direction to time. Here's one reference I can find to that
Huw Price: Philosophy of Time, Boltzmann Brains, and Retrocausality, Robinson's Podcast #58 (Youtube link)
While I don't have the wider knowledge to ascertain exactly what Reichenbach is doing here, it doesn't seem like he is ascribing it to time itself. It seems more of a local method to determine causal order.
An objective directionality to time is a fairly niche position, and for my money the most successful proponents of it use common sense, everyday experience as evidence. For precisely the reason that an objective, encompassing directionality is hard to establish empirically/scientifically. The dominant position seems to be time has no inherent/objective direction; the low entropy big bang suffices to explain all directionality. It would seem out of character for one of the titans of philosophy of science to make the full connection your title does. So I have to assume Reichenbach is being more modest than ascribing directionality to time itself.