Why would a civilisation so incredibly advanced choose a dying star to build such a structure?
IIRC, no age of that Dyson Sphere was mentioned, so I presume it was very, very old and built long before the star was dying.
Like others, I find the idea of a Dyson Sphere fascinating and would have loved to have seen the exploration of it spread across a couple of episodes with or without the talents of a legendary Starfleet engineer to help.
Unfortunately, such amazing discoveries mostly served as a backdrop for the story. You all remember Picard's awe in "The Chase"?
PICARD: I never thought I would see a Kurlan naiskos. Fifth Dynasty? Well, the overall configuration is certainly Fifth Dynasty. The surface ornamentation. Green polychrome over the eyes, and the eyes themselves are closed. This is third Dynasty. From the workshop of the Master of Tarquin Hill.
PICARD: Will, the Master of Tarquin Hill designed ceramic objects that were three hundred years ahead of their time. All we know of him is the work. His name has never been discovered. This object is over twelve thousand years old.
RIKER: The planet Kurl? It's a hell of a long way outside Federation territory.
PICARD: Indeed. I thought your study of Kurlan artefacts was done long ago.
GALEN: I happened to be in the neighbourhood last summer. I couldn't resist. Go ahead.
PICARD: You mean it's complete?
(Picard raises the top half of the pot to reveal a cluster of little pots inside it)
PICARD: Will, the Kurlan civilisation believed that an individual was a community of individuals. Inside us are many voices, each with its own desires, its own style, its own view of the world. The Kurlan civilisation died out thousands of years ago. It is extraordinarily rare to find a figurine intact. Professor, this is an incredible find.
Well, what happened to this priceless you-could-probably-buy-a-planet-with-it relic?
Picard just left it behind in the wreckage of the Enterprise...