Say you submit your application, as planned on June 2nd.
It will take a variable amount of time for verification, probably only a few days if you're in that early.
NO medical schools will see your application until July 1.
After that date, they will receive your primary application, and either screen it or simply send you a secondary for applying. This may take a few days to weeks.
So you will get your secondary applications in early-mid July.
It is considered ideal to have those returned within 2wks, which means that you'd get them in late July.
At that point, some schools will want the LORs before further review, while others will choose interview invites based on secondary+primary and want the LORs a bit later.
So, even assuming that everything goes as quickly as possible, and you return your secondaries very quickly, your LORs will likely not be needed until late July. Your school's timeline is not wholly unreasonable (though if they start that late, I wonder when the last few letters go out from their office?)