Nobody has a good idea how many species there are currently, let alone how many there were in the past and how many of them went extinct. Estimates for the current species count are from 3 to 100 million[0], which is a range too wide to be useful in this context.
Of the mass extinction events in Earth's history, we may be peeking above the baseline rate[1]; the most severe statement we can make is that vertebrate mammals don't appear to be doing very well in recent times, which stands to reason, considering that Homo sapiens is by far the most competitive vertebrate mammal, doing a good job of crowding others out. But there's a lot of life and biodiversity outside of vertebrate mammals.
[0]: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3160336/
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.