Looking for a truly secluded Mediterranean villa with its own private bay/beach — Italy, Greece, Spain?

I’m looking for recommendations for a very specific kind of place for a family trip in July (4 people).
We are NOT looking for:
resorts
hotels
villas near crowded beaches
“beach access”
beach clubs or tourist towns
We are specifically looking for:
a standalone house/villa directly on the sea
its own private bay / cove / effectively private beach
very isolated surroundings
little to no nearby neighbors
natural Mediterranean scenery
The vibe we’re looking for is similar to places like:
Villa Lalli House in Cefalù, Sicily

So far I’ve researched:
Gulf of Orosei in Sardinia
Ogliastra / Foxi Manna area
Meganisi and Lefkada in Greece

But many “private beach” listings still seem to be near other villas or tourist areas.
What I’m really trying to find is something more remote — hidden coves, rocky coastline, turquoise water, almost no people around.
Budget is flexible for the right place.
Would love recommendations for:
specific villas/houses
local agencies
Airbnb/Vrbo listings
lesser-known Mediterranean regions