Remember this is a textual issue. The textual committees for the NA and UBS tests selected Βηθζαθά (Bethzatha) for the most likely text. This is not as simple as studying the geography. I tend to leave an answer like this to the textual experts. They study things such as does the variation match a copyist error? or does it look like the copyist tried to correct an error? All this keeping track of the ages of the texts and their parent texts.
5:2 Βηθζαθά {C}
Of the several variant readings, Βηθσαϊδά has strong attestation but is suspect as an assimilation to the town of Bethsaida on the Sea of Galilee, mentioned in 1:44. Βηθεσδά, though widely supported, is also suspect as a scribal alteration originally introduced because of its edifying etymology (בֵּית חֶסְדָּא, “House of [Divine] Mercy”). In the opinion of a majority of the Committee the least unsatisfactory reading appears to be Βηθζαθά (א 33 Eusebius), of which Βηζαθά (L ite) and perhaps Βελζεθά (D it(), , ) may be variant spellings.In the opinion of a majority of the Committee the least unsatisfactory reading appears to be Βηθζαθά (א 33 Eusebius), of which Βηζαθά (L ite) and perhaps Βελζεθά (D it(), , ) may be variant spellings. The Copper Scroll discovered at Qumran contains a reference to a pool at Betheshdathayim, which the minority of the Committee interpreted as corroborating the reading Βηθεσδά.
Metzger, B. M., United Bible Societies. (1994). A textual commentary on the Greek New Testament, second edition a companion volume to the United Bible Societies’ Greek New Testament (4th rev. ed.) (pp. 178–179). United Bible Societies.