Hi Alfie!
Is the material breathable?
While looking for a dark colored cotton or linen material, I found a generous sized remnant of a slightly stretchy, knit, soft, thin material to use as a cage cover instead. Maybe a nylon blend, not sure, definitely breathable with the weave. It’s not very attractive, but it does the job as needed.

I’ve secured two layers together with a large button and just a little sewing that allows it to be easily used as a single layer during warmer months, or doubled up in the winter.

here is a photo of the material against the light, stretched out simply by gravity. You can see what it looks like as a single layer, and doubled. It drapes nicely and doesn’t unravel, so I haven’t taken the time to finish the edges. Just the big button as a landmark at the ‘apex’, and a little sewing along the corners that align with the top corners of the cage. The button helps me to drape it over the cage as intended.
It does not filter light completely, but it allows me to have a little light on when they’re sleeping without turning them into night owls.

Drafty air is at least minimized.
Maybe that helps with your project.
Happy 2023!
Laura