You're right, using a desktop CPU for a mediacenter is not a good idea.
Laptop CPUs are, however, really well optimised for very low consumption when idling and that's what you should be using for a mediacenter.
Also, IMHO, ASRock Deskmini would not be a good idea for a media center since it's using desktop CPUs. The idle power draw will not be as good as mini pc with a laptop CPU.