Is there any insight into how Starlink solved cooling? One 'expert' insisted that there is no reason to expect that data center satellites would generate any more heat than starlinks.

Those have a power budget of about 1 rack. I would expect a datacenter satellite to need more cooling if it has more compute.

So, most of the power that Starlink satellites use go into the comms, right? Blasting out electromagnetic radiation to receiver stations on earth, and also the laser(?) backhaul between satellites.

Modulo some efficiency losses, most of the electricity it generates is leaving the satellite. Contrast with a datacenter, where most of the energy is spent heating up the chips, and the rest is spent moving the heat away from those chips.