When talking about proxies I mostly see people use them for rendering code in the client or common code which can be used for both, but are usually called in the main mod class nowadays instead of the CommonProxy. Even Choonster's test mod's DedicatedServerProxy is only used for handling errors were something is called on the wrong side. My question is ultimately what you would put in this class that would not work in common or client code.

The only thing I can think of is code that references the world folder. Because the world folder is on the dedicated server when there is one, any code that references it should probably be in the server proxy. But honestly, I'm just speculating and haven't tried any such code, so it might work in the common proxy as well.

Everything in the net.minecraft.server.dedicated package is server-side only, so if you want to access something from there you would need a server proxy.