NO! Is the goal to get into medical school or to get into medical school QUICKLY?
Even if someone thinks that they could possibly produce a well-written, highly-polished AMCAS in just a few weeks, it takes 2-4 weeks to be verified, and it might take a schools 1-3 weeks to send at secondary. Again, you are thinking what is the latest an application can be, where it should be what is the most optimal? Frankly, anyone who does not have their MCAT score by January of application, they are already losing advantage. why? You need MCAT score and GPA in order to target schools. You need to target schools in order to research them, review their previous secondaries, develop themes, and then apply these themes to AMCAS. By the tome this OP got their app complete, they would be late at the top 1/4-1/3 of schools (highly competitive). Invites will be going out in August or earlier with interviews starting then as well.
Why do I do this? of 100% of applicants, 60% get rejected, 20% get a single acceptance, and 20% get more than one. More telling is of 800,000+ applications, the 150 med school programs can at the absolute most, offer 150,000 interview invites. With each applicant putting in 16 apps, on average 3 applications get action. In short, at any individual school, at least 80% of applications must be rejected prior to interview. Lastly, 60% of matriculants take a gap year.
Pay heed to what I say