We don't really have that for many present day medicines. We know they work, statistically speaking. But we don't know for all of them how they work, what the mechanism is. For some drugs this lags the discovery of the drug itself (historically: all drugs, but in the last 100 years fewer of them). For some drugs it simply hasn't happened.
For math, you don't really have empirical data though. There is no trial you can do to find out whether, statistically, the proof works as intended. Because pure theoretical math rarely produces a measurable result.
This is absolutely not true. The reason nearly every mathematician believes the Riemann hypothesis is true is because out of zillions of discovered non-trivial zeros, so far 100% have been located at (0.5, <prime>). If we had only discovered 3 such points, nobody would give it much credence.
There is a difference between formulating a hypothesis based on observations and treating it as true. As I understood it, things that are not proven yet will not be used to formally prove other things.