Extremely unlikely through a tour operator, most of their accommodation is on the cheaper side at Perissa & Kamari. The caldera view properties don't need tour operators, they get booked up anyway as the view is the main reason many people go, as well as it being a popular honeymoon choice.
Doing it independently is really easy if you are set on this place. It also gives you the freedom of combining a few nights here with a beach choice too. Plenty of help available if you do want to book independently. I have just come back from a Peloponnese road trip, & apart from our first accommodation booked it was a case of find somewhere as we went along!
Many people love Santorini & are happy to spend their time on the balcony looking at the caldera, but that becomes too boring for some after the novelty wears off. The time of year can have a bearing on this as a choice to, it can be very windy at times.