In fact, we may embed

Let us first suppose that g_s is not elliptic. Choose an embedding of G into GL_n to reduce to (ℂ). Then by our assumption, g_s has an eigenvalue of norm greater than one, let λ be the case where G=GLabsolute value of such an eigenvalue. Suppose for want of contradiction that the conjugacy class of g_s contained an elliptic element a in its closure. WLOG a is in the special unitary group SU_n. Let h be in the conjugacy class of g_s. Then h has an eigenvalue of absolute value λ. Letting v be an eigenvector, we see that |(h-a)v| is at least (all relevant notions such as λ-1)|v|, so |h-a|≥λ-1, a contradiction.

Now suppose that g_s is elliptic. We may replace G by the Jordan decomposition and centraliser of g_s is G, which is also reductive. So WLOG, g_s is central in G. Now the notion Zariski closure of ellipticality are inherited from the GL_n case)group generated by g_u is a one-dimensional unipotent subgroup of G. Let E be a non-zero element in its lie algebra. This is a nilpotent element. Then by the Jacobson-Morozov theorem, where you say you have already proven this factwe can extend E to a sl₂ triple E,F,H in Lie(G). Now consider conjugation by elements of the form exp(tH) with t real. This shows that g_s is in the closure of the conjugacy class of g, and we're done.

For g in G, write g=g_sg_u as its Jordan decomposition into semisimple and unipotent parts. I claim that the closure of the conjugacy class of g contains an elliptic element if and only if g_s is elliptic.

In fact, we may embed G into GL_n to reduce to the case where G=GL_n (all relevant notions such as the Jordan decomposition and the notion of ellipticality are inherited from the GL_n case), where you say you have already proven this fact.