Understanding the shell rheology of ultrasound contrast agent microbubbles is vital for anticipating their bioeffects in clinical practice. Past studies using sophisticated acoustic and optical techniques have made enormous progress in this direction, enabling the development of shell models that adequately reproduce the nonlinear behaviour of the coated microbubble under acoustic excitation. However, there have also been puzzling discrepancies and missing physical explanations for the dependency of shell viscosity on the equilibrium bubble radius, which demands further experimental investigations. In this study, we aim to unravel the cause of such behaviour by performing a refined characterisation of the shell viscosity. We use ultra-high-speed microscopy imaging, optical trapping and wide-field fluorescence to accurately record the individual microbubble response upon ultrasound driving across a range of bubble sizes. An advanced model of bubble dynamics is validated and employed to infer the shell viscosity of single bubbles from their radial time evolution. The resulting values reveal a prominent variability of the shell viscosity of about an order of magnitude and no dependency on the bubble size, which is contrary to previous studies. We find that the method called bubble spectroscopy, which has been used extensively in the past to determine the shell viscosity, is highly sensitive to methodology inaccuracies, and we demonstrate through analytical arguments that the previously reported unphysical trends are an artifact of these biases. We also show the importance of correct bubble sizing, as errors in this aspect can also lead to unphysical trends in shell viscosity, when estimated through a nonlinear fitting from the time response of the bubble.