I think the blackbird is showing you what the problem is - vine weevils as you suspect. It's like candy to them. You may also have a problem with drainage. I wouldn't bin it just yet. Try the 'spa treatment'. Take that plant up, clean the root ball completely, give it a bath in a gallon bucket of water with a 1/4 cup of bleach. This will kill the weevils & get rid of the eggs & whatever else is in there. Repot with new medium. Try using a cactus mix - a loose, sandy fast draining gritty medium. The stuff you have it planted in now could be wrong for this plant. I suspect it holds too much water - which is perfect for grubs, mold & pathogens. Use a cactus mix and be sure the pot has sufficient drainage. Sedums are succulents and prefer an arid position with lots of sun. They are cousins of cacti. I have Sedum Autumn Joy.

Thank you Bathgate - we have three other Sedum - Autumn Joy being one. All of those are in similar pots with the same compost and they are looking gorgeously healthy, which is why I moved the White one well away. The white one is the newest of them all.

I could have been guilty of overwatering as we have had a very dry period of weather which is unusual, but because I watered all of the Sedums together I couldn't automatically blame that.

Looking closely at the picture, you can see the vine weevil grubs.
Sedums are very sensitive to Bleach so I would be very careful about that. Better to discard the root ball all together and treat the stems as cuttings. Make sure there are no grubs up inside the stems.
Spread out the compost somewhere for the birds to go through.
When the stems have rooted then pot up in a very gritty mix of non-peat based compost and sharp sand or Horticultural grit.

Yes agree with owdboggy, take some cuttings, they are so simple to do, even the leaves will root, put five or six leaves in a pot of decent compost, push them half down and each one will root and form a new plant it’s that simple, don’t over water them.

I still think there is hope for the plant. Cuttings may work as an alternative. A very weak solution of bleach (Hydrogen Peroxide is even better but I don't think it's as accessible in the UK as it is for me here in the US). I've saved many plants with HP. Agree about the sharp sand and grit - - that's cactus mix!

Yes sedums are very sensitive in my experience. I would take cuttings and as Owd says make sure there are no grubs inside the stems, easy to check. Use a gritty compost if you can as it will deter the female vine weevil.

Sprinkle the compost onto your garden soil/bird table so the birds can feed on the grubs.

You also have to destroy the eggs as well and the pathogens they carry which can be on your cuttings but difficult to see. Another alternative is Isopropyl alcohol. Wipe down the whole plant with a paper towel dampened with Isopropyl alcohol. Vinegar solution could also work.

The eggs are laid just below the soil so the stems used for the cuttings wont have eggs on them.

I seem to remember reading in some scientific journal that when there are grubs already in the soil a female wont lay any eggs there. The thinking behind that is, if there are already grubs then there wont be enough food for any extra grubs. It makes sense for the female to find a different plant.

We don't even know for sure what the problem is

Thank you so much to everyone!

I shall try taking cuttings after checking for grubs, and give the soil up to the wild part of the garden for the blackbird, who is busy making her nest today so she'll appreciate them for sure.

Oh that's really nice you have a nest. It's fun to see the chicks grow up and fly off. I have a pair of mockingbirds roosting in my tree. They come back every year and will actually let me approach their nest. Anyone else will get dive bombed. Your blackbird is exactly that same bird as The American Robin - except for the coloring. It's the same bird with all the same antics.

Sorry! Different shape, different size, different family!
It is the American equivalent of the English Blackbird.

They are both in the true Thrush Family and have very similar attributes; beak, brown head, eating habits, droopy wings, eye ring, posture, overall shape, migratory tendencies. Turdus migratorius and Turdus merula - closely related. I'm not an expert, I'm only pointing out that they are plainly in the same family. You don't look exactly like your sister do you? But there's enough of a similarity to see you are related. I don't want to get scientific here Sue. It's only an observation.

The relationship is a very distant one.(both belong to the very large thrush family) But they are both well loved birds - it doesn't really matter much after all! But it is the American counterpart of our blackbird as far as shape and song (and keenness on worms!)go.