Anton's construction depends continuously on the curve but does not seem quite canonical, in the sense that smoothings of two congruent curves may not be congruent. We can ensure that this will be the case by refining the argument as follows.

Given a Jordan curve $J$ in $\textbf{R}^2$, let $C$ be the smallest circle which encloses it, and $C_\epsilon$ be the outer parallel circle at some fixed distance $\epsilon>0$. Then the region between $J$ and $C_\epsilon$ forms a topological annulus $A$. There exists a unique annulus $A'$ of the form $r\leq |z|\leq 1$ which is conformal to $A$ (e.g. see Thm 2.7.3 in Conformal Maps and Geometry by D. Beliaev). Let $f\colon A'\to A$ be a conformal map. Restricting $f$ to circles $|z|=t$ yields a homotopy between $J$ and $C_\epsilon$ which depends only on the congruence class of $J$ (the choice of $f$ does not matter because the only conformal maps $A'\to A'$ are rotations, or an inversion).

**Addendum (8/8/2023):** Igor Belegradek and I just finished a paper where we show that the space of Jordan domains in any surface of constant curvature admits a strong deformation retraction onto the space of round disks, and this deformation is equivariant under isometries of the surface. The proof is based on a point selection theorem from the interior of domains which allows us to apply Riemann's mapping theorem, or more precisely Caratheodory's kernel theorem, to a family of domains, as discussed in a related post.