Political power is when people act in concert, when leaders have people behind them because those people agree with how they're being lead etc., it's not something that happens unilaterally. You can physically force people, but for what when you can have robots that you don't have to force?
So that kind of leaves wiping out nearly all humans, for a few hundred thousand remaining people to get served by robots and play golf. Which I would say is what we're kinda on track for. But that wouldn't be a race between nations, that race would just be pretend to get the tech online; it would rather be a race between all humans and a bunch of sick people.
AI doesn't have to be used offensively to give a nation power. The machines can already design new software at the drop of the hat, it will at some point become capable of designing new consumer products and industrial products. Putting a tireless intelligence at the task of looking for new optimizations in your industrial processes and factories will eventually give you economic dominance, one way or another.
Offensive uses don't need to be as extreme as you imagine them. We're already starting to see the effects of having meticulous automated hackers.
Finally, political power itself doesn't need to come from people acting in concert. There's always been wheeling and dealing and intrigue and bureaucracy. Once AI is sufficiently versed in legal matters, it'd allow political interest groups to overwhelm governments and private enterprises with objections and appeals, paralyzing their ability to carry out their agenda until they make concessions.
The possibilities of using intelligence for acquiring more power are literally endless..
The particular form of the pursuit of power practiced by OAI and Antrophic - the creation of trillion dollar companies, is one of the most benign in human history. It's so unnaturally benign, that I doubt it's going to survive for much longer.