Don't overthink it, book Iliou Villa.
My wife and I had the absolute best stay at Iliou's on our honeymoon. She was a wonderful host from the moment we landed in Santorini until the moment we left. She arranged transportation from the airport, brought our luggage upstairs, provided a delicious breakfast every morning, cleaned frequently, and provided wonderful suggestions for restaurants and things to do in Pyrgos (in particular Pyrgos Restaurant and Venetsanos Winery).
We would love to visit again!