Our society is build on a foundation of relatively cheap energy. Massive amounts of energy are built into the production of every item of food that reaches your fridge or item that reaches your closet. The unleashing of stored energy was the dawn of the Industrial Revolution.
If energy became 1000 or 1 million times cheaper, it would transform society. Global warming would be solved. Food would be solved. Water would be solved. Equal access to all materials goods would be solved. I’d personally be living in a Dr. Evil base on top of an alpine glacier that I myself re-froze. That or a floating city over the South Pacific.
None of this is any sort of explanation for "destabilizing society".
Even if nuclear fusion was perfected today it would take decades to build out and infrastructure would still be a big cost. That would take care of only electricity, not overall transportation.
I agree on AGI, but a massive step function shift in energy will certainly destabilize society. The livelihoods of at least a hundred million people depends on fossil fuels.