Does timing of PhD application submission matter?
I doubt that it matters in a formal sense at very many institutions unless the school used "rolling admissions". However, it is possible that if you apply early and there is some flaw or lack in your application that someone, even a clerk, will catch it and give you a chance to correct it. 

Sometimes a department head will try to shepherd early applications since it is his/her responsibility to make sure that there are enough applications to build a new cohort. That is separate from formal acceptance, however. 

It is also possible that an especially "juicy" application will be passed to a professor who might be looking for a certain kind of background. But again, that is separate from acceptance. 



In rolling admissions, students are accepted or not as the applications come in. The deadline may mean only that the process stops at that date.