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Welcome Super Gran. Silver birches give dappled shade and are quite light and airy. I don't think they would block out the light completely.
What sort of soil do you have? Most magnolias like soil a bit acid. I have alkali soil and I planted magnolia Leonard Messel which is very similar to stellata with pale pink flowers and will grow in an alkali soil. But it's a bit strange that you already have a stellata that's OK. They hate drying out in summer (unlikely last year!), they don't like cold winds and spring frosts can damage the flowers.
If you go for birch I would suggest making sure they are one of the very narrow varieties. I have some, but have forgotten the name! Katsura makes a very nice subject as well - there's something elegant about it.