Of OE spyrd Bowsorth-Toller says,
The word glosses stadium (1) with the meaning a course :-- Ða ðe in spyrde iornaþ qui in stadio currunt, Rtl. 5, 33. (2) with the meaning a measure of distance :-- Swelce spyrdas fífténe (spyrdum fífténum, Lind.) quasi stadiis quindecim, Jn. Skt. Rush. 11, 18. Swelce spyrdo fífe and twoegentig quasi stadia .xxv., 6, 19. Ðara spyrda stadiorum, Lk. Skt. Lind. Rush. 24, 13. In all these passages the West-Saxon uses furlang. [Goth. spaurds (1) a course; (2) a distance: O. H. Ger. spurt stadium.]
It is not clear whether the 'racecourse' sense derives from the 'distance' sense or vice versa; the same is true of the Latin word it glosses, stadium, although the Online Etymology Dictionary suggests the Greek original of the Latin term suggests that the 'distance' sense was prior.
This is a very shaky foundation upon which to build an origin for ModE sport—especially since I find no evidence that the OE term survived into ME.
As OP points out, Middle English Dictionary gives sport(e with the senses:
(a) Amusement, entertainment; pleasure, fun; also, an activity that brings pleasure or amusement; a pastime or game; also, ?a sexual exploit, an amorous deed [quot. ?c1450, 2nd]; don sportes, to play games; haven (taken) ~, take (one's) pleasure, have fun; ?participate in merrymaking; maken ~, create amusement, make sport; (b) a source of pleasure or delight; (c) joking; foolery; in ~, in jest; connen no ~, to engage in no foolery.
Solace, consolation; also, ?a means of comfort or consolation; maken ~, to console (sb.), cheer up.
There are also related words, sportaunce, sportelet, sporten, sportful, sporting.
MED sees all of these as "Shortened form[s] of disport", "disporten, &c., which first appear in ME a generation earlier than sport(e and its relatives. For the noun MED gives the following senses:
(a) An activity that offers amusement, pleasure, or relaxation; entertainment, merry-making, fun, recreation; maken ~, to entertain (sb.); taken ~, amuse oneself, have fun; (b) a pastime, sport, or game; also, the game of love, flirtation; (c) in ~, in jest.
(a) Pleasure taken in an activity or enjoyment derived from it; haven ~, to take pleasure (in sth.), be gratified; (b) consolation, solace; a source of comfort; don ~, to cheer (sb.) up.
(a) Deportment, conduct; customary behavior, custom, manner; (b) an instance of behavior, an act or activity; don ~, to do something.
Departure; maken ~, to set out (for a place).
The first two of these senses are clearly identical with those of sport(e. They carry over into EME, whence they give rise to the modern senses.
Among the "disportes" mentioned by the MED citations are dice, reveling, minstrels singing songs and telling jests, and finding Venus on a bed of gold, as well as recreations which would be regarded as "sports" today, hawking, hunting, angling, archery.
None of the citations alludes to racing or reflects (except for one allusion to the "actes and disportes Olimpicalle") a sense of "competitive" sport.
And there is no other MED headword of the form sp?rt*, sp?rd*, spr?t*, or spr?d* which could be taken as derivative of spyrd.
It looks like the similarity of the OE term is coincidental, since it cannot be traced into ME.
If I may ...
The origin of the word gold itself can be traced, via German, to the Gothic gulth.
http://www.dictionary.com/browse/gold?s=t
That said, the original Dutch for "guilder" (a golden coin, as well as the official currency of Netherlands until 2002 A.D. is gulden. It was also common in many German territories.
Which would suggest that John Walker may have been wrong when he said
It is much to be regretted that the second sound of this word is grown
much more frequent than the first
in assuming that "the second sound" was a new development rather than the old version that, through a quirk of fate, had regained some of its erstwhile popularity. That is if we wish to trust him at all in this case. He may have thought that the sound had "grown much more frequent" whereas in reality it may have been frequent all along. 18th Century experts are hardly infallible.
Best Answer
What Happened?
Lich in Old and Middle English referred to bodies in general. In the OED's two definitions for lich, that body could be alive ("lich, n.," def. 1) or dead ("lich, n.," def. 2). That's important to understand because corpse could also refer to a living or dead body ("corpse, n.," def. 1 and 2).
Between Middle English and Early Modern English, two changes occurred:
When Did Lich Decline?
Corpse (in earlier forms like cors) was in Middle English by the 13th century, according to the Middle English Dictionary, and stays in the language from then on:
Lich, in contrast, is swiftly disappearing by the 16th century, appearing primarily in printed editions of older manuscripts, like this (printed 1600 from a text dated to around 1350):
The usage may have stuck around in spoken language for much longer, but usages in print after this point are intentionally archaic, like this 19th century ballad:
Why does this replacement occur?
One guess is that speakers favored the more clinical, educated sound of the French-derived cors/corpse over the common use of lich. Such lexical substitution was common in the Middle English period due to the influence of England's Norman aristocracy (Wikipedia).
Another is that lich had an uphill battle because it looked the same as a very common word, like. The lists of forms for lich and like look rather alike; both could appear as lic, lich(e), lych(e), lyke, and like. They may have sounded the same too. So it's also possible that lich underwent a decline in usage compared to a similar-meaning word (corpse) in order to avoid confusion. At around the same time, corpse specialized to fill its current semantic niche.