I'm fairly certain that he's not meant to be a closeted gay man, but a psychopath who will pretend to be gay and sleep with someone he isn't interested in, in order to get what he wants.

I think this is right. The Willicks are the embodiment of the maxim that "everything is about sex, except sex, which is about power." They are both the most sexually prolific characters in the show, and at the same time, the characters least motivated by sex. It's entirely about power for them.