Those MCAT dates would definitely not prevent you from getting your apps in early IF you're organized and on top of everything. Even though AMCAS opens for submission June 1st, no apps get sent on to medical schools until the end of June, and you can submit AMCAS without an MCAT score (you'd just put in the date you expect to take the exam, and then once you take it, the score would automatically get uploaded).
The month of June gives AMCAS time to process all the applications they receive. So theoretically, you could submit your AMCAS app June 1st, get verified sometime in mid-June, and then your application would be ready/available to the school(s) you submitted to when your MCAT score is ready in early July (which would be only a few days after the first apps get sent at the end of June). Most of the time when people do this, they only submit to one school (usually their state school), so that if they end up bombing the MCAT, they've only officially applied to one school and can withdraw without losing all the application $ and becoming a reapplicant everywhere. Another benefit to that would be the ability to finalize your school list once you've seen your actual score (though hopefully you'd have taken lots of practice tests and know what score to expect).
HOWEVER, the plan above would require you to have your AMCAS app all ready to go by early June while you're still preparing for the MCAT. It's easy to underestimate how much time AMCAS will take to fill out--it's not just the personal statement! So I would say it would definitely be doable but only if you're organized and confident that you'll be able to study for the MCAT and work on AMCAS at the same time. If you think worrying about AMCAS would detract from MCAT studying, it's not worth it, especially if you're banking on a great MCAT score.