Here’s a format that worked very well:
My friend ran and hosted a book group for several years. Having a consistent location was a definite plus. Some of us hate driving around alone in the dark looking for a strange address.
One person would present the book, giving an overview, perhaps some background on the author or setting, and a few starter questions: five to ten minutes, tops. Then the discussion would proceed, prompted as needed by the presenter with the list of questions.
At the end of two hours, there’d be about half an hour of general socializing, and then we ended.
My friend would gather all book suggestions at the end of the year for the next year’s worth of meetings and then present them by e-mail. People would speak for their choices. The person who proposed a given book was expected to be the presenter. Not everyone could host, but anyone could present.
I’m sure my friend quietly arbitrated among choices and maybe favored some over others, but she just delivered a list of selections—no voting, no vehement crusading, no bitter arguments back and forth. If there was a book you couldn’t stand, you could just skip that meeting.
If someone who’d picked a book actually dropped out, we went ahead and read the book anyway, and someone would volunteer to present.
This combination of structure and flexibility worked very well and avoided competition and contentiousness.
…
You don’t owe someone else the right to define your notion of fairness. And you have not committed the group to following some rigid policy of majority rule. Not every situation lends itself to a strict democratic process, and in fact in a small group a voting structure makes it very easy for a few to dominate and consistently shut others out. A situation of this sort needs a leader and not a ballot box. In my opinion.
You’ve already set the structure, and one noisy whiner doesn’t oblige you to change it. I’d say that if she’s so unhappy with the format and the way you conduct the meetings, perhaps she’d be more comfortable in another discussion group.