I e-submitted mid to late june. submitted my DAT scores mid july. Got my last letter in late august. 3 interview invites by late september.
All e-submit does is queue up your application to be reviewed. It ensures you have the minimum/imperative details filled out (personal info/GPA/coursework) so by the time whatever schools you applied to take a look at your credentials they know to return to your application for further review upon recieving other information (post graduation grades/transcripts, additional LoR's, etc).