Do bottlebrush trees shed their bark?

Question

The bark is almost foam feeling.

Answer 1

I believe you must have quite an old specimen!
The lifespan is 20 to 40 years for this tree.

The peeling bark is a normal process, though yours appears to be a bit excessive.
I do see some leaf spot virus on the leaves.

You could treat for this.

Apply sulfur sprays or copper-based fungicides weekly at first sign of disease to prevent its spread. These organic fungicides will not kill leaf spot, but prevent the spores from germinating.