If you have a mediocre application, your chances of an acceptance are higher if you apply earlier in the cycle. If you have a strong application, you'd have a decent chance of an acceptance even if you wait until the deadline, but possibly not at an institution as competitive as what your stats would suggest is your niche. So it's incorrect to think that you'd have no chance by applying late in the cycle, it's just no going to be your best chance.
You can moderate the effects of applying later in the cycle by submitting your application to AMCAS 6 weeks before the score will come back to be sure you are verified before the score returns. Then schools have your entire application immediately when the score is distributed. You can also prewrite your Secondary essays (from prompts fround here on SDN) so you can return each Secondary within a day or two of getting the formal invitation.