I can't encourage birds into the garden as I don't think they should feature in an a la carte dinner for the cat!
We have just moved in and wondered if rather than getting rid of it I could plant something, no idea what as it's very shallow, I don't think I want to put a pot on top of it - but will anything grow in such shallow depth
I think you would have a perpetual problem of it being too dry. There's not enough depth for a water reservoir of any kind, and when it does rain repeatedly, there's nowhere for the water to drain away and anything that likes dry conditions will rot.
You might be able to get a succulent going, or one of the ferns that seeds itself into walls, but I still think it will struggle.
A bowl with drainage holes would work very well, and you could have two or three and swop them through the year - bulbs followed by summer flowers followed by autumn/winter planting.
Succulents should work. I saw on a program the other week they shoved succulents into a wall they'd built with random bits of rock so was a few nooks and crevices. Soil wasn't deep. Obviously no idea if they lasted or died but it was Charlie Dimmock who did it.
Sempervivums would probably work quite well. They are very succulent (and hardy), don't need much depth, and would spread to fill the space. I vaguely remember my great-gran had some S. arachnoideum (the cobweb houseleek) planted in a bird bath. They used to be planted on roofs, supposedly to protect from lightning, so are as tough as old boots.