One of my former professors in my undergraduate university, who is the chair of China and Taiwan Studies, talked about this in a class on three major turning points in modern Chinese social history (Boxer Uprising, 1911 Revolution, Cultural Revolution). She is no fan of Mao Zedong, to put it lightly, and grew up during the Cultural Revolution, so she knows its horrors in a pretty personal way. However, she also said that as far as accomplishing what it set out to do, destroy the olds and bring in the new, it was absolutely successful. There is no doubt that the CR was a major player in the destruction of traditional Chinese culture. However, in its place, she has said, the People's Republic of China has not really stumbled upon a real identity; instead, in its modern state, it is a society obsessed with consumerism, not unlike the US. And, actually, one of the best ways to understand this change is to compare modern mainland China with Taiwan (which is essentially the representative of “traditional Chinese culture” in the modern era), which obviously did not experience the CR and in fact basically did the opposite, going to some lengths to preserve many traditions.
The anthropologist Robert Weller, who specializes in Chinese culture, and particularly religion and politics in Taiwan, has written a lot about how traditional Chinese culture has facilitated the growth of democracy in Taiwan. In making this argument, he discusses quite a bit about how Taiwanese people continue to adhere to traditional Chinese belief systems that were all but eradicated by the end of the CR. These include the “ethnic religion”, as well as Daoism, Confucianism, and Buddhism. Temple and shrine worship, and religion in the home, are a part of everyday life in Taiwan in a way that hasn’t been the case in the PRC in decades (though it is beginning to recover a bit now). Further, over the decades the PRC has made destruction of these elements, along with the eradication of all non-Mandarin Chinese languages, a part of its policy, even down to the writing system. The direct enforcement of some of the policies, but not the language one, has waned in recent years, but they still do tie in with the PRC’s push to create a homogeneous Chinese identity on the mainland. Taiwan has not done this. This is simply not the reality of Chinese culture or ethnicity, which is actually incredibly diverse. In the PRC, Confucianism is essentially dead, mainly used as a prop by the government as a way to guilt citizens into believing authoritarian rule is justified; outside of this usage by the state the philosophy is no longer particularly significant, according to the professor I mentioned. This is, again, not the case in Taiwan. That’s not to say Taiwan hasn’t changed; it’s just that it has been more “organic”, you could say. Taiwan is very gender equal, obviously not an aspect of traditional culture, whether in China or most other places in the world, really. However, one could argue that this is just a growth in the philosophy of Taiwan and how it is applied to modern issues. Philosophy is not static after all. But what we don’t see is a situation like in the PRC, where a war was conducted against the traditions. Taiwan has embraced its "Chineseness", so to speak, and adapted the culture to modern views in the country. What was done on the mainland was a destruction of philosophies and traditions, not an adaptation of or engagement with them, and in fact the mainland really takes part in this sort of partial rejection of "Chineseness" while still claiming parts of it when convenient. I think it could be argued that the PRC has a very different and new way of being "Chinese" compared to Taiwan, which has developed a very different identity.
My sources were mostly the talks from that professor I mentioned, as well as some from Weller’s Alternate Civilities: Chinese Culture and the Prospects For Democracy and Melissa Brown's Is Taiwan Chinese?.