If you apply the first week of June, it takes a few days to get your transcripts verified. And yes, AMCAS'll start this without the MCAT score. If you ait to submit your Primary application until the first week of July, it will take 3-4 weeks to get verified because the waiting line is so long. So you'll be complete about early August.
If you have an outstanding application, you'll be fine if you get your Secondaries back within a few days (though it takes most a month to get ddone once the flood begins). If, however your application is weak in any way, you'll be best off if your application is complete sooner, in which case you should consider the pre-MCAT submission plan. At worst you'd be out $130 if you apply to one school and find out that your MCAT is so low that you decide not to add more schools.