Both empirical and theoretical investigations of scale-free network models have found that large degrees in a network exert an outsized impact on its structure. However, the tools used to infer the tail behavior of degree distributions in scale-free networks often lack a strong theoretical foundation. In this paper, we introduce a new framework for analyzing the asymptotic distribution of estimators for degree tail indices in scale-free inhomogeneous random graphs. Our framework leverages the relationship between the large weights and large degrees of Norros-Reittu and Chung-Lu random graphs. In particular, we determine a rate for the number of nodes $k(n) \rightarrow \infty$ such that for all $i = 1, \dots, k(n)$, the node with the $i$-th largest weight will have the $i$-th largest degree with high probability. Such alignment of upper-order statistics is then employed to establish the asymptotic normality of three different tail index estimators based on the upper degrees. These results suggest potential applications of the framework to threshold selection and goodness-of-fit testing in scale-free networks, issues that have long challenged the network science community.