Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence of the word "countably" by "finitely" there (both in the statement and the proof), I'm not able to observe where the proof fails. Maybe it's some silly mistake that I overlooked, but it also maybe some mistake one easily commits without even realizing it and leads to a wrong conclusion. Anyway, thanks in advance for this seemingly naive question.

In the last paragraph of Kaplansky's proof, the construction could yield an infinite number of nonzero $x_{ij}$, even if each $M_i$ is finitely generated. The infinite matrix he produces will have finitely many nonzero entries in each row, but could have infinitely many nonzero rows. 

