I know it isn't true because our flight was delayed due to storms last summer and we arrived in Orlando around 2:30 a.m. and believe me....there was absolutely not a single person at the ME desk or any ME people anywhere nor busses. About the only people at the airport were cleaning folks. The whole section where ME is located was dark.
I pulled out my ME info and called the toll free number and thankfully someone answered. She authorized a taxi cab to take us to the hotel (and pay for it - we paid tip) but DH has to fish out our luggage cause they had put it in a locked room since it was designated for ME.
I would call ME and ask what happens if your flight is delayed and arrives in the middle of the night just to make sure what to do. I have yet to read of anyone on this board who had a flight come in so late and ME was there waiting for them.