Majority of them may not eat wildlife meat but a looot of them take Chinese medicine which have these ingredients. And that's a cause of illegal wildlife trade.
I remember there is a time when the government try to lower the consumption of shark fin a few years ago, what they manage to do is that they stopped the legal importing (it is legal to import shark fin with restriction and licenses, but most airline inbound to Hong Kong will not take the cargo if there are shark fin present) but what it lost in legal import, the illegal import increased to compensate. You just make it easier (now that they don't go though legal mean) and more expensive (now that its harder to come by)
The problem with exotic food in Hong Kong and Asia is like the problem with drug in South America. You have a large demand on those item, and you have people that are poor enough to take the risk, which mean no matter how much you try, you most likely ended up losing, and as Chinese in general got richer, again, as long as there are Chinese that live below poverty line, these trade will still be there because there are market, and there are people who can pay for it to sustain the market.
It's hard to next to impossible to alter the eating habit of a person, IMO, you probably better off have better control of wet market than banning it, it's like Chicken trade, you centralise the butchering centre to make it safe to human consumption. If these trade goes underground, this would mean the government would be 10 times harder to control it.
The problem is that China has 1.4 billion people. So even if 1% acts like shithead, you are talking about at least 14 million people which is a big market still. If China was the size of Thailand, all this wouldn't matter.
Most of the world may not like American foreign policy, but they still admire the US and aspire to things American.
You can't say the same about China. Ask most people around the world what they think of when they think China, it's repressive evil government, crappy cheap products (even though this is changing), weird people eating disgusting things. I guess it doesn't matter to you like it doesn't matter to most Chinese what others think because they lack social skills and situational awareness.
Compare that to S. Korea, which has 1/30th of China's population. S. Korea is loved by a lot of people around the world because of its tremendous soft power. China is pathetic in comparison.
These people you answer your post to is the reason why China have zero soft power. That's because they don't care. And you cannot not care what other people think or what other country think until you are big enough and more importantly, influential enough to sway what people think, and China have not reach the point yet. And if at this stage China appear to have ignore the worldview like what these people think, the world will see China as a sense of overinflated ego.
What can increase China soft power? Children. Not that children are the future, but if you control a generation of kids via TV Program or Toy, you control the view of the whole generation. Why kids? That's because kid's mind are simple, and when these kids growth up, they are going to impart their own value into their own kids, so they basically help to control the next generation of kids as well. There is a reason why Disney publish all these superhero movie, and there is a reason why toys like GI Joe is popular not only in the US but in the world. What they basically trying to do is to groom a generation of kids and young adult into Americanised Value. If China want to control the world like the US? They need to come up with the Kiddy movie of the century and toy of the century. At this moment, even Japan did it better with Anime and Computer games than China.