Let \( {{\bf R}_{\mathbb S^{d-1}}}(p\to q) \) denote the best constant for the \( L^p(\mathbb R^d)\to L^q(\mathbb S^{d-1}) \) Fourier restriction inequality to the unit sphere \( \mathbb S^{d-1} \), and let \( {\bf R}_{\mathbb S^{d-1}} (p\to q;\textup{rad}) \) denote the corresponding constant for radial functions. We investigate the asymptotic behavior of the operator norms \( {{\bf R}_{\mathbb S^{d-1}}}(p\to q) \) and \( {\bf R}_{\mathbb S^{d-1}} (p\to q;\textup{rad}) \) as the dimension \( d \) tends to infinity.  We further establish a dimension-free endpoint Stein-Tomas inequality for radial functions, together with the corresponding estimate for general functions which we prove with an \( O(d^{1/2}) \) dependence. Our methods rely on a uniform two-sided refinement of Stempak's asymptotic \( L^p \) estimate of Bessel functions. This talk is based on joint work with Błażej Wróbel.