I chanced on the following explanation on Aug. 5 2018, 2 years and 5 months after I initially posted this. Anyone have anything to add to or buttress it?
The origin of fast meaning quickly, rapidly etc. was already present in OE and appears to derive from a Scandinavian usage:
The meaning "quickly, swiftly, rapidly" was perhaps in Old English, certainly by c. 1200, probably from or developed under influence of Old Norse fast "firmly, fast."
This sense developed, apparently in Scandinavian, from that of "firmly, strongly, vigorously" (to run hard means the same as to run fast; also compare fast asleep, also compare Old Norse drekka fast "to drink hard," telja fast "to give (someone) a severe lesson").
Or perhaps from the notion of a runner who "sticks" close to whatever he is chasing (compare Old Danish fast "much, swiftly, at once, near to, almost," and sense evolution of German fix "fast, fixed; fast, quick, nimble," from Latin fixus).