The MCAT and application stuff on top of your normal premed 80 hour workweek is indeed bad. I specifically took easy courses that semester on purpose. I put about 20 hours a week into MCAT prep on top of everything else. Was definitely working 90 hours a week considering everything, I had no life whatsoever.
It's your decision. You can absolutely do this, but you're going to be battening down the hatches and not seeing your family/friends much for the next 6 months. You just have to decide if you want to kill yourself studying like that or not.
This is why gap years are so common now. Shadowing, volunteering, clinical experience, getting published, and working to pay for stuff if you don't have rich parents, on top of getting a 4.0 GPA and a competitive MCAT score, is VERY hard to do in 3 years. I did it, but I was working just as hard back then as I am now in medical school.
Applying in June isn't ideal but it's fine. But don't push it any later, just take a gap year at that point.
Edit: don't forget to budget time for the CASPER and SJT as well. Part of the reason I hate those tests so much is that, during spring of your application year, you have too much going on anyway. I was applying the first year they came out.....
