My understanding is that the client-side isn't generally the bottleneck for MMOs, it's all the server-side computation that has to happen to make it all run. Calculating behaviors, physics, etc for hundreds or thousands of players simultaneously is really hard, and Google Stadia doesn't change that in any meaningful way.

Edit: Well, now that I think about it, it depends entirely on the game; PlanetSide 2 is an example I can think of off the top of my head that brought my computer to its knees during big battles. But most MMOs aren't exactly pushing the envelope of graphics quality or physics simulation, so it's not as big of an issue.