In my conclusion (so far):

Blood flow is important for cooling the brain (as well as the core of the body).

In many non-human mammals and birds there's evidence for a mechanism that allows the brain to be cooler than the rest of the body. This mechanism, called SBC (selective brain cooling) has actually several flavors, but all seems to involve just head-level "radiators" like the nose. 

[![enter image description here][1]][1]
[![enter image description here][2]][2]

There was no direct evidence (as of 2004) that SBC exists/works in humans, although some speculation based exist based anatomical considerations.

Info and images from 

* Michał Caputa "Selective brain cooling: a multiple regulatory mechanism", 
https://doi.org/10.1016/j.jtherbio.2004.08.079


  [1]: https://i.sstatic.net/SvNp2.png
  [2]: https://i.sstatic.net/BPSkA.png