The complexity of matrix multiplication is a central topic in computer science. While the focus has traditionally been on exact algorithms, a long line of literature also considers randomized algorithms, which return an approximate solution in faster time. In this work, we adopt a unifying perspective that frames these randomized algorithms in terms of mean estimation. Using it, we first give refined analyses of classical algorithms based on random walks by Cohen-Lewis (`99), and based on sketching by Sarlós (`06) and Drineas-Kannan-Mahoney (`06). We then propose an improvement on Cohen-Lewis that yields a single classical algorithm that is faster than all the other approaches, if we assume no use of (exact) fast matrix multiplication as a subroutine. Second, we demonstrate a quantum speedup on top of these algorithms by using the recent quantum multivariate mean estimation algorithm by Cornelissen-Hamoudi-Jerbi (`22).