DP -> HDMI conversion hardware is pretty intense. I haven't owned a device that does this and doesn't at least get warm.
It can be, but it doesn't have to be. DisplayPort is mostly a superset of HDMI, with different voltage levels. Going from HDMI to DisplayPort is much harder than going from DisplayPort to HDMI, though the hardware to do this is apparently much cheaper than it used to be, which is nice to see.
DisplayPort to HDMI only requires level shifting, as I understand it. I'm sure someone will correct me if they see this and I'm wrong.
But also yes, most display port implementations also speak hdmi and can get away with simple converters
You might be thinking of DisplayPort++, which is a special port supporting both DisplayPort and HDMI, allowing passive adapters. This is not available in the USB-C DisplayPort AltMode.