It's important to recognize that, as AGI enables other disruptive technologies, nobody can really predict how things are going to work out. That's kind of the premise of the Singularity.

An AGI somehow figuring out post-scarcity economics and genuine democracy certainly could come to pass - but I expect it would have to be an ASI and I'm not smart enough to suggest how it would work.

In the near term I'd expect AGI to facilitate more technical solutions like large scale carbon capture, advanced materials science for batteries and renewables and dramatically better and cleaner energy sources. Advances like these could solve the issues you brought up without requiring full-scale economic/social restructuring.