To avoid an opinion-based answer, one should focus on the first paper on the subject, (1858), by Arthur Cayley, "A Memoir on the Theory of Matrices", Philos Trans 148, preceding Hamilton's 1864 paper (strictly on quaternions!).
In it, Cayley, a major intuitive genius of the 19th century, basically invents and "takes ownership of" matrices. At the bottom of the very first page, he gloats about his
"[...] remarkable theorem, that any matrix whatever satisfies an algebraic equation of its own order, the coefficient of the highest power being unity, and those of the other powers functions of the terms of the matrix, the last coefficient being in fact the determinant; the rule of the formation of this equation may be stated in the following condensed form, which will be intelligible after a perusal of the memoir, viz. the determinant, formed by the matrix diminished by the matrix considered as a single quantity involving the matrix unity, will be equal to zero."
In his superb pedagogical way, in paragraphs 21, 22, 23, he states it and illustrates it for 2x2 and 3x3 matrices, and adumbrates the logic and the concepts involved in the general case.
A Cayley and his intellectual twin, J Sylvester, (like Campbell the discoverer, 1897, of the CBHP composition of matrix exponentials), were "practical" geniuses who sought the "point", the answer, instead of hidebound, inhibited and often counterproductive continental chapter-and-proof proper mathematese. They were communicating in the more old-fashioned 18th century language, which only survived in the Soviet Union in the 20th century. I don't see a priorities' drama unfolding here, but I am aware of the comical priorities' battles waged on the misnamed "BCH" formula mentioned.