I'm no expert, but it's my impression that the faction that ended up controlling Taiwan is not the "culture of old China". It's the westernized elite that was trying to reform China as a democracy.
Taiwan became a democracy because because the people of Taiwan asked for it and the regime at the time, the Kuomintang, wished to seek democratic legitimacy in response to these demands. Since then, Taiwan has grown to become one of the most socially liberal societies in Asia.