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The etymology of the word [経緯]{いきさつ}, in the sense of "the details of how something came to be" seems completely opaque.
Even on the Japanese internet, the only reference I can find is this article opining that it derives from an expression [行き]{いき}[世話]{せわ}[体]{てい}, without citing any references. This etymology sounds quite unlikely to me given the strange reduction せわてい -> さつ.
Does anyone with better sources than me have a plausible etymology for this word?
Shōgakukan's 日本【にほん】国語【こくご】大【だい】辞典【じてん】 is usually a pretty good reference for etymologies. However, their entry for 経緯 as visible here at Kotobank doesn't provide much detail about where the いきさつ reading came from. Here's what we can tell just from the entry:
This is a relatively recent word, with the first citation given from 1779.
The entry includes no historical kana spellings for the いきさつ reading (compare for the けいい reading further above on that page), indicating that this is not related to 域【ゐき】.
The definition for the いきさつ reading doesn't include anything about textiles. I suspect any "textile" connection is purely from the spelling, which looks to me like it's probably 熟字訓【じゅくじくん】, where the kanji and the reading don't have a direct derivational relation.
The dictionary editors parse this as comprising two morphophonemic (sound-meaning) components, as a compound of いき + さつ.
Let's look at these pieces.
The definition itself includes the phrases 「事のなり[ゆ]{●}[き]{●}。[ゆ]{●}[き]{●}がかり」, suggesting that the initial いき may be the 連用形【れんようけい】 or stem of verb 行【い】く. This seems reasonably straightforward: "how things have gone" is one way of translating the concept of いきさつ.
But what about that さつ?
There are no obvious good matches for this that I can find in any reference. No kanji with this reading seems to have any meanings that align usefully: さつ kanji at Weblio, さつ kanji at WWWJDIC.
As @jogloran notes in the question itself, the linked blog post at https://ameblo.jp/gogen3000/entry-12542274891.html makes the unsupported claim that いきさつ is from 行【い】き世話【せわ】体【てい】, but the clear mismatch between せわてい and さつ renders this extremely unlikely on phonological grounds. The author also explains this as 「進行する世話の体（テイ）」, but I cannot find any examples of such a phrase 行【い】き世話【せわ】.
⇒ Considering the phonology, the relatively recent appearance of the term (making radical phonological changes harder to account for), and the apparent lack of evidence for the component phrase, I think we must rule this out.
A different page I found online at https://boutex.jp/archives/2245 just makes the vague claim that, apparently because いきさつ is spelled with the kanji 経緯, the reading いきさつ must have something to do with textiles and latitude and longitude. But this connection is left unexplained and unreferenced. The post author appears to be unaware of, or to be deliberately ignoring, the possibility of this being 熟字訓【じゅくじくん】.
⇒ Considering the lack of any explanation for how the reading いきさつ connects to the concepts of textiles, or latitude and longitude, I think we must rule this one out as well.
Speculation
Given the mid-Edo-period appearance of this term in popular literature (specifically, 洒落本【しゃれぼん】, a genre focused on humor in the red-light districts), it is possible that this term may have arisen from slang. I wonder if いきさつ may have evolved from wordplay? Considering that some sources define いきさつ as 「込【こ】み入【い】った事情【じじょう】」 ("complicated circumstances"), one possibility might be from the term 行【ゆ】き雑【ま】じり ("going and getting all mixed together") → ゆき shifts to いき, 雑【ま】じり changes to the 音【おん】読【よ】み reading of ざつ and then shifts to さつ.
The source of the term いきさつ has been lost to history, as best I can find in resources currently available to me. The initial いき is probably from 行【い】き, but the derivation of the さつ part remains mysterious.
