We study the satisfiability threshold and solution-space geometry of random constraint satisfaction problems defined over uniquely extendable (UE) constraints. Motivated by a conjecture of Connamacher and Molloy, we consider random $k$-ary UE-SAT instances in which each constraint function is drawn, according to a certain distribution $π$, from a specified subset of uniquely extendable constraints over an $r$-spin set. We introduce a flexible model $H_n(π,k,m)$ that allows arbitrary distributions $π$ on constraint types, encompassing both random linear systems and previously studied UE-SAT models. Our main result determines the satisfiability threshold for a wide family of distributions $π$. Under natural reducibility or symmetry conditions on $\operatorname{supp}(π)$, we prove that the satisfiability threshold of $H_n(π,k,m)$ coincides with the classical $k$-XORSAT threshold.