Now come on, What's with the terrible beach comments for Anassa? Sure, it's not the Maldives, but it's a big, dark sandy beach with virtually nobody else on it and the sandcastles work fine. You will certainty not find any of your criteria on mainland Europe.
This is a truly child-friendly place but it is also very, very stylish for us grown-ups. Outdoors, there is a child-only pool, an adult pool, and indoors, there is a big pool in the really good spa.
The scenery is magic. They have a buffet in one of the restaurants, and they also have some of the best suites or adjoining doubles you will find in Europe. It is MUCH cheaper than Cap-Ferrat (isn't everywhere?).
Do not dismiss FS Provence, as although it is inland, it is brilliant for kids. September is a great time to go.
Europe does not do beaches+sunshine. And if we do, it's not well.
Cannot imagine that Villla Milocer would suit.