The qualitative intuition that you mention — that $\mathbb{R}^n$ begin to be similar in sufficiently high dimension — is the same intuition underlying my question Does the truth of any statement of real matrix algebra stabilize in sufficiently high dimensions? from some time ago.
Specifically, I had asked whether every statement in the language of what I called real-matrix algebra eventually stabilizes in $\mathbb{R}^n$ for sufficiently high dimension. The language has sorts for scalars, matrices, row vectors and column vectores and operations for the vector space operations. But the language does allow one to make explicit reference to the dimension. I had thought that statements in this language might stabilize in high dimension.
But the answers showed that that was wrong, and that one can find statements whose truth values do not stabilize in high dimension, but rather detect various number-theoretic features of $n$.
Thus, those answers tend to refute the intuition that even for very simple statements, $\mathbb{R}^n$ begins to look alike in all sufficiently large dimension.