When I actively rented, my absolute policy was that until I had money in hand, there was no rental and I would continue to list. I didn't do reservations. The reservation goes to the first person who produces money.
If they whined that there were other sites that would allow them to make a reservation and pay on arrival, I would tell them to book at one of those sites. Of course, the those sites (hotels.com, booking,com, expedia.com) charged about twice my rates.