I probably view it too pragmatic but to me, the F statistic is merely a result of a calculation that is used to (magically) determine the value that I'm really interested in: the p. APA, though, wants the F reported along with the degrees of freedom (and the p). Why is knowing the F and the df relevant? Is this used as a short test whether my calculation was right? Or is there any information in the F value that I don't get?
I can't speak for the APA but I see value as a reader in having information on degrees of freedom and F statistics. Degrees of freedom provide reassurance about the scale of the study. The F statistic, as the ratio of explained variance to error variance, contains important information about how effective the model was.
This cuts both ways. With a significant $p$ value based on a high F but few degrees of freedom, I might worry about issues specific to a particular data set and a spurious or non-generalizable result. With low F but many degrees of freedom, I might wonder if a statistically significant result really is substantial enough to consider significant for practical application.
So please report these as a courtesy to your readers even if a journal doesn't require them.