From the little I've read and watched regarding animal intelligence, it's typically mentioned that the brain-to-body size ratio is more important/correlated to estimated intelligence than brain size alone. It seems to help explain things like the outsized intelligence of certain smaller animals (e.g. corvids), but it also seems to only be a partial explanation.
Makes more sense to dedicate larger share of body-mass to brains the smarter the brain is. So I don't think it is that the big brain makes them smart, its that smart brains can make up for being a bigger burden so they grow larger.
What matters though is how efficient were the H. Sapiens brains.