I'm not in any position to tell committees how they should run their process. There are other factors, such as the availability of faculty or other interviewers that dictate the timeline, as well as other policies, such as the test score requirement. As the chair of our prehealth committee, I can tell you nothing frustrated me more than taking hours to craft an evaluation letter only to find out later after sending out the letters that the person bombed their exam (MCAT, DAT, whatever) and thus withdrew from the process. Thus, it's probably in this committee's memory that the tendency of the student advisees is to take MCAT's late, so they have no choice for the purposes of making most efficient use of their time but to wait until scores are in to do interviews.
I'm certain that most admissions committees though are aware of when letters are sent out from particular schools, and from their side, they'll be able to adjust the consideration of people from your school until those letters get sent in. They also know well if those letters really contribute significantly to their overall decision for people who they interview before getting the letter or packet. Would the admissions committees want those packets to be in earlier? Of course they would, but they also know the advisors writing those packets well enough to know that timeline.
In short, I wish you lots of luck, but just imagine what the considerations are to changing the timeline to the people who run the committee and make additional suggestions to help adjust the administrative timeline so that letters could be sent out. You may find out from talking to the committee members and the chair that it's much more difficult than you anticipate. Or (better) that they may be thinking about it anyway.
That said, I know that there is a tendency that more schools are going into an earlier timeline to interview candidates. Used to be, one would have the first set of interviews scheduled around Labor Day or thereafter, but obviously now a few schools have first interviews in mid-August. Most of the people in these pools are going to be the out-of-this-world-stats applicants who applied very early and know they're going to command attention in the process that I don't think committee letters would make a huge impact when it comes to getting an interview. So, maybe everything's working just fine from the standpoint of really making the pitch to the applicants who really need convincing and persuasive letters and packets.