60 day notice to vacate at lease end and turning in keys after last payment. (MI)

Hey all. So my current lease is up on July 27th. I bought a house back in May and gave them my 60 day notice to vacate before my lease is up, as they required. We are moving into the home next week.

I completely understand that im on the hook for rent in July. I told them I would prefer to pay my rent for July when its due then turn in my keys. The lady at the leasing office told me i have to turn in my keys on the 27th or else I will be subject to early lease termination fees. They did a pre move out inspection already and stressed that they want a quick turnaround for my unit since its due for renovations. Am I missing something here? They get my rent for the month of July even though im not living in the unit. It seems like it would be easier for everyone if I turn in my keys the same day i pay my final rent payment. I dont have to drive back out here to drop off my keys and they can start renovation earlier than expected.