That's about as clean as old rush seats can get ... the fibers are dust collectors. Removing the varnish would probably destroy the rushes. They are just grass leaves, twisted.

If they get wet, they don;t dry well and fall apart and mold. Ugly to see.

You could re-do the seats with fresh rush or other cord. It is not difficult, just tedious.

When I donated some furniture to The Salvation Army, they refused to take anything with wicker or rush materials. They don't want known dust-catchers.

I just replaced the rush fiber seat in an old Ethan Allen dining chair. There are excellent internet instructions and even a Youtube on doing it. Ordered rush fiber (kraft paper rush) from Amazon. I was quite pleased with the results. Will coat the seat with shellac. I am an old basketmaker and knitter so don't know if that experience helped. Peppygirl