We examine the role of local linear instabilities in entropy-stable discretizations within the summation-by-parts (SBP) framework. Local linear instability, as introduced in [Gassner et. al., 2022], refers to the linearized discrete operator exhibiting perturbation growth exceeding that of the corresponding continuous problem. We argue that linear instabilities should be interpreted as a secondary mechanism of numerical error, whose practical impact must be assessed relative to more dominant error sources. For high-order split-form discretizations of the Burgers equation, we derive bounds on perturbation growth consistent with the continuous problem, thereby establishing convergence for smooth solutions using the arguments of [Strang, 1964]. To assess the practical significance of local linear instabilities, we employ Floquet analysis to account for the time-varying dynamics of the linearized operator. Numerical experiments suggest that local linear instabilities introduce negligible error compared to both physical transient perturbation growth and to the numerical error associated with baseflow propagation. We further show that local linear instabilities in entropy-stable SBP schemes are strongly amplified by the presence of simultaneous approximation terms, with unstable modes being highly localized at block interfaces. Interface dissipation therefore suppresses the associated perturbation growth. We also use both theoretical arguments and numerical evidence to demonstrate that any remaining local linear instabilities from volume terms are effectively eliminated by including volume dissipation. Finally, we investigate local linear instabilities for more general systems, including the compressible Euler equations. We show that the apparent local linear stability advantages of central schemes observed for the Burgers equation do not persist in broader settings. Overall, we conclude that local linear instabilities do not pose a practical obstacle to the use of entropy-stable methods.