What's the difference, then, between a Leihbücherei and a Bibliothek? I thought a Bücherei was basically a bookstore (for selling books) and a Bibliothek was basically a library (for borrowing books).
The earliest lending libraries were not public but private, funded by subscriptions, like a club. Women probably did discuss the latest bestsellers when they met at social occasions, but it wasn't a circle or group designed specifically to meet for discussion; I would have thought that that was a very late-20th-century concept. (Oprah Winfrey.)
Lending libraries might have been funded more by women than men because women had much less access to higher education, so they had to take the initiative if they wanted information. Limiting membership to women was also probably an issue of propriety; in polite society, ladies did not attend mixed social events except when chaperoned by men, so it had to be clear that nothing improper could be going on there. Another related factor might be that in the 19th century, when the novel became a popular genre, the whole idea of fiction was still considered rather scandalous, or at least frivolous, and in practice, many novels were written for, and read mainly by, women.