TL;DR: Similar usage is much older than paper checks. But the rumor is not far from the truth, especially if the question is about the U.S. dollar currency specifically.
The usage inside of (modern) English texts per se (I mean especially inside full sentences written down) is predated by standalone, symbolic usage in English speaking environments. In this sense it is comparable to a question like "why do the Arabic numerals list digits with the most significant digit first" as far as the applicability of the English tag goes.
To illustrate that claim, the once very influential book by Titus Bennett "A New System Of Practical Arithmetic Particularly Calculated For The Use Of Schools In The United States - Containing A Large Proportion Of Examples In Federal Money In Each Rule Throughout The Work" was still using syntax like "1 dol. 25 cts." throughout its several editions during the first half of the 19th century.
The reportedly earliest preserved use of the dollar sign on an American financial document (while the symbol itself predates the existence of the U.S.) is displayed here. As you can see there, both the "textual form" and the "symbolic form" of the amount on the bond are written so as to make it difficult to prepend or append any further digits to the amount illicitly, exactly as the rumor stipulates they should, without dwelling on the check vs. bond distinction. The textual form spells out even the numerals in full. The symbolic version of the amount reads "$185&98cts." Of course they could have perhaps written something like "=185$&98cts." instead, thus better matching the equivalent textual form, but then they'd have to include a symbol meaning nothing in particular just to have something to put immediately to the left of the leftmost digit; quite a few other languages did exactly that on financial documents in the paper-and-pen times.
Neither the dollar symbol itself, nor similar compact usage on American financial instruments were in no way new by 1792 (although the United States dollar currency, initially pegged at 1:1 rate to the then nearly worldwide used Spanish dollar which used the same symbol, was brand new.)
This colonial banknote from 1750s, for an earlier example, denotes the amount of 97 pounds, 10 shillings and 0 pence as "£97:10:00."
In old British usage, you could write "£1. 2s. 5d." (pronounced as "one pound, two and five".) There's some additional info that could potentially be used to track that order of symbols back to the 14th century on the Royal Mint website (I don't have Oxford English Dictionary at hand to pursue the particular reference), but it presumably connects the English usage to mediaeval Latin.
This Roman coin includes an inscription "HS NOVIES MILL" in which the thing closest to a conventional currency symbol (namely, "HS") precedes the number, while the normal word order in Latin would have the number precede the "HS". (Nota bene: This particular coin inscription is part of a full sentence.)
Perhaps the usage comes from an even earlier language with a different word order or perhaps it has extra-linguistic origin. It is also equally possible that the word order of "HS" in "HS NOVIES MILL" would determine whether the coin stood for 900,000 sestertii or 900,000,000 sestertii; Latin with its free word order normally depends on word endings to indicate parts of speech and cases, and we lose that for a monetary symbol, therefore the word order could be a trace of the underlying grammar of the sentence. Numismatists don't seem to be sure about this anymore.
Either way, the coin, as well as the British pound-shilling-pence notation, both predate paper checks by centuries.