"tall, broad-leafed grass growing on the margins of streams or in other wet places," Middle English rēd, rede, from Old English hreod "reed, rush," from Proto-Germanic *kreut- "reed" (source also of Old Saxon hraid, Old Frisian hriad, Middle Dutch ried, Dutch riet, Old High German hriot, German Ried), with no known cognates beyond Germanic.
Meaning "musical pipe made from a reed stem" is from late 14c. (reed-pipe is from c. 1300). As part of the mouthpiece of a musical instrument it is attested from 1520s. Meaning "a reed instrument" is from 1838. Figuratively, as a type of frailty, etc., from early 13c.
adjective suffix, "full of or characterized by," from Old English -ig, from Proto-Germanic *-iga- (source also of Dutch, Danish, German -ig, Gothic -egs), from PIE -(i)ko-, adjectival suffix, cognate with elements in Greek -ikos, Latin -icus (see -ic). Originally added to nouns in Old English; used from 13c. with verbs, and by 15c. even with other adjectives (for example crispy).
<a href="https://www.etymonline.com/word/reedy">Etymology of reedy by etymonline</a>
Harper, D. (n.d.). Etymology of reedy. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/reedy