Role of "HDMI Baluns"?
I suspect HDMI to Cat6 converters don't actually contain baluns, because as you point out HDMI TMDS signals are balanced signals just as UTP pairs are balanced.

In the days before HDMI, video signals were generally carried on coaxial cables.  To avoid using expensive coax, many manufacturers came up with a method to use cheaper Cat5 cable instead.  Since coax is unbalanced, and Cat5 is balanced, you must use a balun (a balanced to unbalanced transformer) to connect the two.  A/V technicians became used to this term to mean a device that converts from one cable type to another.

I believe HDMI to Cat6 "Baluns" are simply called that because A/V technicians are familiar with the term.