I stayed at Joy's Rooms Ortigia, but did not take any excursions from them.
Likely, they themselves do not do the excursions and other options they offer, (Cave tours, Boat tours, Rickshaw and tuk-tuk tours) but use local operators they have arrangements with, and get some cut of the booking.
In essence, they are a third party booking agent, but then most of the other places advertising a wide range of tours in Ortigia are as well. On one hand, they are no worse (or better) than any website you will find. As for "trust", yeah, they will take care of you and make sure things go OK. They have an interest in making sure you are happy.
Like others are saying, Noto is easy to get to be public transport, easy direct connection. I am not sure the sights warrant a "must" for a guided tour, if they are offering that. You can get as much by doing it yourself.