Short summary first, long rant follows. Summary: Mathematics and small parts of philosophy and many other disciplines rightly qualify as non-empirical science, but no lesson can or should be drawn from them about empirical science. Empirical questions are fundamentally different from non-empirical questions.

Long rant:

I don't disagree fundemantally but what you say about philosophy irks me as a philosopher. If you looked at a random sample of publications in philosophy you would find out that only a very small percentage of them has the rigour and exactness that we associate with science. I have also met many philosophers, perhaps even the majority of all I've met so far, who would not describe themselves as scientists.

The remaining part of philosophy that adheres to strict standards and uses mathematical method is akin to mathematics and computer science, and perhaps most similar related to formal linguistics and economics in the way it works. I personally consider this part of philosophy a science, it is a kind of applied mathematics, although often speculative and conditional on axioms that are not as evident as in mathematics. Other disciplines have similar non-empirical parts, a typical example is Social Choice theory in sociology, which I would consider a scientific theory, although it is not empirical. In the end, it's applied mathematics.

I agree that mathematics and the relevant rigid and formal parts of other disciplines are non-empirical science, but to these areas the debate about hypothesis testing and the right use of statistics simply doesn't apply. It a fallacy to presume that, because these fields exist, clearly empirical disciplines (or parts thereof) could do without statistics. If a question is empirical, then it has to be addressed with proper quantitative methodology.

This is extremely important to me personally, since I've been in deep disagreement with colleagues for many years about this issue. They work in related disciplines within our philosophy institute and habitually make "qualitative empirical analyses" of texts without treating them as mere precursors to quantitative studies. They see no problems with their methodology, even when I point out to them that the size of their samples would be too small to support the generalizations they make if they did make quantitative studies. To me, this is absolutely crazy, I just can't see how a qualitative analysis of 20 texts could allow you to take these as representative for hundreds of thousands of texts if a quantitative studies of the same texts could not possibly reveal anything useful because the sample size is too small. What's worse, their whole discipline seems to be based on these kind of extremely small scale qualitative empirical studies, plus a very vague mix of fairly imprecise philosophy and common sense. I'm a nice person and get along with my colleagues well, so I won't ever tell them my opinion, but if I'm honest, I'd say that their discipline is pseudo-science or, at best, imprecise, non-scientific philosophy in disguise. (To make this clear, I have no problem with imprecise philosophy and have done it occasionally myself, I just don't think it can qualify as science and not many philosophers consider it as such.)

Long story short, empirical question have to be addressed with quantitative methodology or you get what I'd call "elaborate opinions".