John:
Nothing unique about Santorini regarding "family" churches. Throughout the Balkans, such churches can be found. In the older sections of Berat, Albania, there are numerous family churches. It should be understood that when these churches are opened, they are open to all. Occasionally, they might be located inside of a family compound and so a tourist might not know that there is a service there, but locals would know.
Name days have traditionally been far more important to Orthodox Christians than birthdays. Birthdays were generally not recorded. Baptism & Chrismation were recorded in the church register. So a perfect name day starts with preparation for the Eucharist, receiving the Eucharist at the church of your patron saint, and gathering for a meal with family and friends.
And what is wrong with medieval, the most glorious period for the Eastern Churches? While building a small Church in honor your patron saint to worship God is no guarantee of health, happiness, and long life, it certainly isn't going to hurt. Our God is a jealous God.