Let $\mathfrak{ g}$ be a complex simple finite dimensional Lie algebra and $G$ be the adjoint Lie group with the Lie algebra $\mathfrak{ g}$. To every $C \in G$ one can associate a commutative subalgebra $B(C)$ in the Yangian $Y(\mathfrak{ g})$, which is responsible for the integrals of the (generalized) $XXX$ Heisenberg magnet chain. Using the approach of arXiv:1708.05105, we construct a natural structure of affine crystals on spectra of $B(C)$ in Kirillov-Reshetikhin $Y(\mathfrak{ g})$-modules in type $A$. We conjecture that such a construction exists for arbitrary $\mathfrak{ g}$ and gives Kirillov-Reshetikhin crystals. Our main technical tool is the degeneration of Bethe subalgebras in the Yangian to commutative subalgebras $\mathcal{ A}_\chi^{ \mathrm{ u}}$ in the universal enveloping of the current Lie algebra, $U(\mathfrak{ g}[t])$, which depend on the parameter $\chi$ from the Lie algebra $\mathfrak{ g}$ (and are of independent interest). We show that these subalgebras come from the Feigin-Frenkel center on the critical level as described by Feigin, Frenkel and Toledano Laredo in arXiv:math/0612798. This allows to prove that our affine crystals in type $A$ are indeed Kirillov-Reshetikhin by reducing to the crystal structure on the spectra of inhomogeneous Gaudin model which is already known (arXiv:1708.05105).
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