Think of biodiversity as the entropy source for all of the random number generators that drive biological innovation.

When biodiversity goes down, there are fewer potential biological resources that can be tapped to solve new problems.

You are correct that biodiversity has a natural upwards component from speciation. But the timescale (which you describe as "quickly") is many orders of magnitude slower than the timescale of the downwards component that extinction has. A species that took literally tens of millions of years to speciate and evolve can be wiped out in months.