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.