The Kondo effect describes the scattering of conduction electrons by magnetic impurities, manifesting as an electronic resonance at the Fermi energy with a distinctive temperature evolution. In this Letter, we present a critical evaluation of the current methodology employed to demonstrate Kondo behavior in transport measurements, underscoring the limitations of established theoretical frameworks and the influence of extrinsic broadening. We introduce an approach for analyzing spectroscopic indicators of the Kondo effect, employing the Hurwitz-Fano lineshape as a model for the Kondo resonance in the presence of extrinsic broadening. Through precise scanning tunneling spectroscopy measurements on an exemplary spin-1/2 Kondo system, phenalenyl on Au(111), we demonstrate the efficacy of our proposed protocol in extracting accurate intrinsic Kondo linewidths from finite-temperature measurements. The extracted linewidths exhibit a robust fit with a recently derived expression for the temperature-dependent intrinsic Kondo linewidth, providing compelling evidence for the validity of the underlying theory.