science

The most pressure it would experience would be the difference in internal vs external pressure. At 1000ft of depth there’s a pressure of 440psi. Assuming the sphere somehow managed a perfect vacuum that’s still well below the 6000psi compressive strength of high strength concrete, hell they would still have more flexural strength. The spheres themselves definitely wouldn’t be the weak link.

It seems to me pumping out water to near vacuum while under crushing outside water pressure would be a significant power hog

Well, yeah. That's the point. It's a battery. Whatever energy you put in to pump the water out, you get some percentage (probably in the 50-70% range) of it back when you let the water back in. The point of these is to store energy from renewables whenever they are providing more power than the grid demands - otherwise the power would be wasted.

Edit: The paper claims 72% efficiency which is pretty good if I understand things correctly

I don’t know what depths we’re talking about here,

From the article:

The idea is relatively simple: hollow concrete spheres are installed at a depth of several hundred metres.

... you’d need some massive ...
from the srticle :
... a sphere nine metres in diameter and weighing 400 tonnes will be submerged ...

Can you calculate the weight of a sphere of 9 m of displaced water ?
No ? Well, it is 382 tons.
So, the concrete sphere is already massive by itself. "You" don't need any complicated anchoring.
Same goes with the rest of your mechanical engineering intuitions : you did not work in this domain or study it, did you 😆 ?
Also, stress cycling is bad on most material, yes. But here it is compressive stress and the geometry is symmetric. Without further study, i want to believe this thing has good potential and my intuitions tells me it looks nice. Time will tell 😁 !