I’ve always found this a pleasant lightly humorous phrase, given that it is about education but implies poor spelling.

Is a specific originator for this phrase known?

The OED says “The phrase is said to have originated in a toast proposed c1807 by the English banker and politician Sir William Curtis (1752-1829).”

This attribution is supported by the first citation:

1825 Mirror of Lit. 29 Jan. 75/1 It has been very much the fashion amongst a class of persons to attribute to Sir W. C. certain bulls… He is charged with having given, at public dinners, the following toasts:—‘The British tars of Old England’. ‘A speedy peace, and soon.’.. ‘The three R’s—Reading, Writing, and Rithmetic’.

Interesting: if that’s true, it didn’t originate as a conscious joke but as a “bull” (i.e., “an expression containing a manifest contradiction in terms or involving a ludicrous inconsistency unperceived by the speaker").  Just as Sir William didn’t notice that “soon” didn’t add anything to “speedy,” he didn’t notice that two of the three “Rs” didn’t actually start with the letter r.