It's a valid concern. We use democracy right now to address inequality. Are you concerned that those in charge of the abundance machines will create their own nations with militaries to prevent other nations from taking said resources? To this I would respond: there is nothing stopping other nations from building their own abundance machines. As we've seen with AI, distillation cannot be stopped. The knowledge of how to build these machines cannot be censored. Here in Denmark I have high confidence that people will vote to build nationalised abundance machines for the good of all.
Perhaps this is more of a regional risk? There are some nations which culturally value individuality. I'm not sure the U.S. would build national abundance machines.
I think I broadly agree with your thesis, however. Abundance is going to force massive social change. If goods and services are effectively free, most jobs disappear. So too does their income and consumer demand, which destroys the entire premise of capitalism. You might be surprised to learn that Elon Musk believes capitalism will cease to function in the next 10-20 years because of this. So abundance will force us to move to a system which much more closely resembles communism. It's the transition I'm worried about.
When Middle American manufacturing jobs were offshored to China, Hilary Clinton famously laughed at them and told them to "learn to code." I fear many more people whose jobs are lost to AI will be met with similar derision. It is only when things get very bad that change will occur. The Star Trek timeline describes this collapse between 2020s-2053.