Forests can reach a certain form of stability. However, succession is a thing. What we no longer have much of is old growth forests. Letting forests get old trees creates a particular kind of diversity now rare. Some things only exist in this habitat with large fallen trees.

My assumption is that a diversity of species, age, living/dead wood, dense/sparce tree cover/open ground will promote biodiversity of other species. Trees that die are left where they are. (I take a little for firewood.) I live in a coastal temperate rainforest biome in North Devon, UK.