I'm reading Erhan Çınlar's book on Probability and Stochastics, and in Chapter 2, he says that distributions are used extensively in elementary probability theory in order to avoid measures. And again, in notes to Chapter 2, (in reference to probability distribution) he says "The distribution functions are avoided; as Neveu (1965) noted, they should have disappeared long ago." Why does he say that? (As in, why should one avoid probability distributions?)
Distribution functions (which are not the same as probability distributions) are a tool, just as characteristic functions for example.
Tools should be judged by what they help us accomplish. To prove the Central Limit Theorem, the characteristic function is an invaluable tool. To describe the distribution of the maximum of an i.i.d. sample, the distribution function is very useful (whereas the characteristic function is not).
I am not aware of the context of the quote, but it sounds a bit like telling a painter not to use a particular type of brush. Perhaps the author's intent was to tell the painter that there are other useful brushes, i.e. not to rely exclusively on distribution functions. While it is certainly true that some or even most subjects in probability theory can not even be formulated without using measure theory, this in my personal opinion does not constitute an argument against distribution functions, as using one does not exclude using the other.