The above analysis seems circular: aren't our social interactions more complex in large part because of our intelligence?
I don't think there is a circularity problem. You can sometimes describe the same events in different ways, as in the chicken-egg riddle.
Our social adaptations and intelligence are surely intimately linked, but are you suggesting that social adaptation alone drove the evolution of our intelligence?
Yes, that is precisely what I am suggesting.
It's opinion, of course. The last time I checked, fossils didn't come with IQ reports.
I can readily see how there could be selection for social cohesion. I find it hard to come up with a scenario where there could be selection for intelligence. It is often remarked that the most intelligent produce less children than average. I cannot prove that this was always so, but I do suspect that. The most intelligent are often social misfits.