For $α\in (1,2)$, we study the following stochastic differential equation driven by a non-degenerate symmetric $α$-stable process in $\mathbb{R}^d$: \begin{align*} {\rm d} X_t=b(t,X_t){\mathord{\rm d}} t+σ(t,X_{t-}){\mathord{\rm d}} L_t^{(α)},\ \ X_0 =x \in \mathbb{R}^d, \end{align*} where $b$ belongs to $ L^\infty(\mathbb{R}_+;\mathbf{C}^{-β}(\mathbb{R}^d))$ with some $β\in(0,α-1)$, and $\mathbf{C}^β$ denotes a Besov space (see Definition (2.2) below). The coefficient $σ:\mathbb{R}_+\times \mathbb{R}^d \to \mathbb{R}^d \otimes \mathbb{R}^d$ is a measurable matrix-valued function. The noise $L_t^{(α)}=(L_t^{(α),1},...,L_t^{(α),d})$ consists of independent $1$-dimensional symmetric $α$-stable processes, and is referred to as a cylindrical $α$-stable process. We establish the well-posedness of weak solutions to the SDE, and provide quantitative stability estimates with respect to the drift coefficients.