Thank you for your detailed, informative reply. I'll give a more thorough read soon and write a proper reply.

Thank you for your detailed, informative reply. I'll give a more thorough read soon and write a proper reply.

What's the definition of Q? I don't see it in the original question.

Thanks a lot! For others, I'll remark that $(\cos^2 \theta_1, \ldots \cos \theta_k^2)$ are singular values of the matrix $A_2^\top A_1$, where $A_i$ is a $(n \times k)$ matrix representing an orthonormal basis of $\pi_i$. To see that these angles characterise $(\pi_1, \pi_2)$ up to $O(n)$-action, one can consider another $(\pi_1', \pi_2')$ with the same angles, WLOG assume that $\pi_1 = \pi_1' = \mathbb R^k \subseteq \mathbb R^n$, and do some linear algebra.

The only proof I know is by puncturing each space and calculating relative homology. Is there a simpler proof?

@YoavKallus thanks a lot! "Diameter" is a better phrasing of the problem I posed.

Thanks for the reference Yoav. I think separation in that paper is the minimum of all pairwise distances, while I am considering the maximum of all pairwise distances. Do you know if this is addressed there or elsewhere?