Here’s W. V. O. Quine discussing the fact that Frege didn’t adopt a type theoretic approach (like Russell and Whitehead’s) when faced with Russell’s paradox.
Actually, it is not to be wondered that Frege did not think of this course, or, thinking of it, adopt it. It was by having all his classes at ground level that he was able to avoid the use of high-level attributes, and this he liked to do. (“On Frege’s Way Out”, Mind, vol. 64, 1955, p. 489)
Now I have no idea whether Frege thought of effecting his reduction of arithmetic to logic in a type theoretic setting, but I suspect that if he had, his reasons for not doing so were better than “I like to avoid high-level attributes”.