The issue is not people exploiting one another to rise above each other. The issue is a few people exploiting AI's they own to keep themselves above everyone en masse.
They don't lose the need to rise above you, rather staying above you is the entire point of the AIs they're creating. If you want to compete in the economy in the future, you will have no choice but to also exploit their AIs to try to rise above others. Thereby helping the owning class rise further above you.
Why not make your own? Because you don't happen to have an army of AI researchers to create it for you. Why not use open source models? Again, you can, but you're betting those models will be better than commercial models. And right now the capability gap there is widening.
People in power wouldn't fight AIs to keep themselves in power. Rather you, yourself, using the AIs they own to make a living, would more deeply entrench their already extant power.
Just in case someone didn’t catch the reference: https://xkcd.com/1053/