The usefulness of the graph is much reduced when there are many curves visible, since they are often on top of each other. That's why we require an extra click to show everything in the graph. Also attachment keys in the graph are not very interesting to see, since they are just points in time without any vertical component. I know your use case is to move keys, but we have to think about the more typical usage of the graph, which is primary to visualize your curves.
For an operation to move all your keys, I think the dopesheet is the best choice.
That said, we could have a hotkey to show everything in the graph. I'm not sure that would be much better for your use case, since it's still an extra action to remember before moving all keys (click or hotkey).
