What or who is the source of the proscription on contractions in formal writing?

I followed a bit of a rabbit hole from Stuart F's useful comment. It's probably not possible to find a single target to blame, but I've narrowed it down to one event and one influential author in the 1700's:

• Prior to the Proceedings in Courts of Justice Act 1730, law in England was written in Latin, Norman French, and sometimes English. However, that English was actually “court hand”, which was abbreviated to the point of near illegibility, and total illegibility to non-lawyers, making laws of the era “more difficult to read than Latin records of preceding ages” (per Hendrickson 1971). That act requires that law be “not in any hand commonly called court hand, and in words at length and not abbreviated”. It's doubtful that contractions as we now understand them were among its targets, but they were certainly among its victims in English law. This act is no longer in force, but English and Commonwealth law to this day is completely free of contractions and abbreviation. That proscription may have leaked out of law schools and been imagined as relevant to “formal” language more generally.
• Jonathan Swift (or as I call him, J'ift) abhorred contractions: “... condemn those barbarous mutilations of vowels and syllables. In this last point the usual pretence is, that they spell as they speak; a noble standard for language! to depend upon the caprice of every coxcomb, who, because words are the clothing of our thoughts, cuts them out, and shapes them as he pleases...” (The Prose Works of Jonathan Swift, Volume IV). His logic amounts to little more than sneering at the lower classes, but he was doubtless an influential writer of his era, and his words echo still. He was a linguistic prescriptivist more generally (English's first?), wishing “that some Method should be thought on for ascertaining and fixing our Language for ever” (A Proposal for Correcting, Improving and Ascertaining the English Tongue). These sources were compiled in Julie Axendra Bishop's 1998 thesis, but the commentary is my own.

There are doubtless others of the same era with similar puritanical views, but Swift's influence and law's force suggests that they're of particular importance. Beyond that, this is essentially a meme; people continue to repeat it as authoritative in spite of all authorities disagreeing.