Is the thermodynamic stability against reduction/oxidation of an electrolyte dependent on the electrode material?
Technically you are correct. The thermodynamic potential of electrolyte decomposition should not be affected by the electrode composition.


  which just influences kinetics, not thermodynamics.


The word "just" here is doing way more work than you're giving it credit for. There are situation where the kinetics of electrolyte decomposition are so slow that there is effectively no reaction. The stability window of electrolytes commonly used in Lithium-ion batteries, for example, is smaller than the potentials needed to charge the materials. However, once the reaction begins, initial decomposition forms an inter-phasic layer on the anode known as the solide-electrolyte-interphase (SEI). The kinetics of decomposition on this SEI are slow, which allows the battery to be operated at high potentials[1]. In fact, damage to the SEI has been linked to capacity loss in Li-Ion batteries over extended cycling[2].