Let's dive into this etymology.
(My reference, unless otherwise stated, is Shogakukan's 国語大辞典. I've got a dead-tree copy, and there's also a decent online version available for free via Kotobank. Note that Kotobank's layout is a bit confusing for terms spelled with kanji that have multiple readings.)
Sense development
さようなら
Listed here as first appearing in texts from around 1742. Derived as a shortening of さようならば.
さようならば
Listed here as first appearing in texts from around the late 1500s, early 1600s, as a conjunction. The meaning was as expected from its parts: [さよう]{like that, that way} + [ならば]{if it be}. The "goodbye" sense appears later, cited to a text from 1791.
→ We can surmise that the "goodbye" sense for either form probably showed up in the early 1700s.
Derivation of the parts
さよう
First cited in the 伊勢【いせ】物語【ものがたり】 (The Tales of Ise) of the early 900s.
This is a compound of さ ("that", medial distal marker, cognate with modern そ) + よう ("way, manner, appearance").
- さ
First appears in the Nara period (710–794) with the form さて. The shorter さ then appears from the Heian period (794–1185). Modern そう was formerly さう, and this さ may be the first component of that.
Commonly spelled 然 in references. While cognate with the そ in words like そこ・それ・そなた etc., the そ version was used for physical and concrete things, while さ was used for abstract and psychological things.
Note that the character 然 is borrowed from Chinese, but the reading sa is purely Japanese. 然 in Middle Chinese was read as something like //ȵiᴇn// instead, resulting in the modern Japanese on'yomi of zen.
- よう
Commonly spelled 様 in references. This is the only component of this phrase that derives from Chinese, specifically from earlier glyph form 樣 ("appearance; form; look").
[In]{ } Middle Chinese when this was first borrowed into Japanese, the pronunciation was probably something like //jɨɐŋ// (the "j" here represents a sound like "y" in English; reconstructed based on linguistic research into sound changes and sound correspondences). The historical kana spelling in Japanese is やう, representing how it was probably first pronounced in Japanese: //jau//. (We see the same //au// → //oː// shift here as we saw above with さう.)
ならば
The term さよう is a 形容動詞【けいようどうし】 or "-na adjective". The -na modifier particle in modern Japanese comes from older attributive (noun-modifying) なる, which comes from a contraction of even older に + ある. This なる developed some time in the Heian period when 形容動詞【けいようどうし】 were first used.
[The]{ } ならば in さようならば, the oldest form of this term, is this older なる for -na adjectives, conjugated into the conditional or hypothetical form.
- In modern Japanese, the conditional / hypoethetical is created by adding a ば onto the conditional / hypothetical verb stem, which ends in -e (sometimes with an extra -r- in the middle). Consider 行【い】く → 行けば, or 食【た】べる → 食【た】べれば. This -e stem evolved from the ancient / classical 已然形【いぜんけい】 or "realis conjugation" which expressed something that is or could be real, used in subjunctive or suppositional constructions.
In ancient Japanese, the conditional / hypothetical could be formed using the -e stem, or also by using the -a stem -- also called the 未然形【みぜんけい】 or "irrealis conjugation" which expressed something that isn't real. This is the same as the modern -a verb stem used for negatives.
([There]{ } seems to have been some subtlety of meaning differentiating the -eba usage from the -aba usage, but I haven't yet read a fuller exploration of that topic.)
Update: I've learned a bit more about classical Japanese since I wrote the strikethrough portion above. 😄
In classical usage, the -eba ending was a conditional, a bit like English "because". This uses the classical 已然形【いぜんけい】 or "realis conjugation", talking about something that has happened or is assumed will happen.
- 「XXXす[れ]{●}[ば]{●}、YYY...」 means "because you're doing XXX, then YYY".
- 「XXXへ行[け]{●}[ば]{●}、YYY...」 means "because you're going to XXX, then YYY".
Meanwhile, the -aba ending was a hypothetical, a bit like English "if". This uses the classical 未然形【みぜんけい】 or "irrealis conjugation", talking about something that hasn't happened (maybe not yet, maybe never will).
- 「XXX[せ]{●}[ば]{●}、YYY」 means "if you do XXX, then YYY".
- 「XXXへ行[か]{●}[ば]{●}、YYY」 means "if you go to XXX, then YYY".
Conclusion
The only piece with any Chinese derivation is 様【よう】. The best interpretation of the etymology given in A Word A Day is that it was somewhat scrambled, and the "ultimately from Chinese" portion wound up in the wrong place in the sentence.
Addendum
Part of my family background is from Minnesota. When I was first learning Japanese and was taught the background meaning of さようなら, I realized it basically parses out to "well, if that's the way it's going to be" -- which somehow seems like a really Minnesotan way of saying "goodbye".
Please comment if the above does not address your question, and I can edit to update.