You wrote about mushrooms growing at the base of trees being a bad sign that your tree is in danger. We have two birch trees with moss growing at the base. Is this also a bad sign, and should we pull it up?
Please do not panic. Mushrooms on a tree's root require an evaluation, but the moss around your birch is not a problem, just a symptom of certain soil conditions, including wet, underfertilized, compacted or highly acidic soil. Shade also contributes to the growth of moss.
I like the woodsy look of moss. It seems to belong there. Since you have a birch that tolerates soil on the damp side, you don't have to worry too much about that as you would with other species of trees.
If you don't like moss, moss killers are out there, but it will likely regrow while conditions remain favorable to it.