However, we have ATP Synthase. That’s also an ion driven molecular rotary engine. Of course, we use it to convert ionic gradients to energy (ATP).
Both are exceptionally efficient molecules for their tasks.
I'm not sure that adds up: It's not that much longer - 500 million years out of ~3.5 billion?
And the evidence shows that Eukaryotes have evolved much faster and further - just look at the Eukaryota domain. And IIRC part of the reason is advantages such as 'sexual' [0] reproduction - rather than just splitting and (cloning?) DNA, combining chromosomes from two parents.
[0] If that's the correct term.
But the first complex multicellular life only appears about 1.6 BYA, and sexual reproduction and cell differentiation only about 1.2 BYA.
And no, I wouldn’t say eukaryotes have evolved faster or further, just differently. It’s questionable they’d be able to do any of this without the mitochondrial and chloroplast endosymbiotic events, and microbiomes have existed since the beginning as well. Framing it as a contest misses the point, in the end.
But the thing is that, AIUI, ATP-synthase is in the mitochondria and chloroplasts, so from an evolutionary perspective it is in prokaryotes... just symbiotic ones...
In some ways, it’s wrong to imagine the branches of life are actually branches. Loki archea and alphaproteobacter together gave us the last eukaryotic common ancestor, and then bacteria continued to live around and in eukaryotes.
That said, mitochondria can’t be called prokaryotes within us. They’ve given up too many genes, and can’t survive on their own.
Of course, the rest of us can’t survive without them either, so it’s a complete mutual situation.