The Gauss linking integral of two disjoint loops in $S^3$ does not depend on the stereographic pole

Let $\gamma_1,\gamma_2:\mathbb{R}\to S^3\subset\mathbb{R}^4$ be $C^2$, $1$-periodic loops with disjoint images. Let $N\in S^3$ be a pole missing both loops, and suppose the Gauss linking integral of the stereographic images,

equals the integer $n$. Then $n$ is the linking number of $\gamma_1,\gamma_2$ in the sense of IsLinkingNumber: the integral equals $n$ for every pole $N'\in S^3$ that misses both loops.

Proof idea: the Gauss integral is constant along any $C^2$ homotopy of the pair of loops that keeps them disjoint (its $\tau$-derivative is the integral of an exact form on the torus, because $u\mapsto u/|u|^3$ is divergence free). Moving the pole along a path $N_\tau$ in $S^3\setminus(\gamma_1\cup\gamma_2)$ and composing with rotations $R_\tau\in SO(4)$, $R_\tau N_\tau=N$, turns the change of pole into such a homotopy, since the integrand is invariant under rotations. The complement of two $C^1$ curves in $S^3$ is path connected: the great-circle cones over the curves from a point are images of $2$-dimensional sets, so they have measure zero and a midpoint avoiding them exists.