First of all, UChicago is a top-6 school, with a great grad program.
The other places you've mentioned are also good places for doing graduate studies, with many graduates getting NSF postdoc grants and getting into good places.
Second, some Chinese schools are known in the West (definitely Tsinghua and Peking, I personally also know some faculty in Fudan and Sichuan). You need to understand that someone from a lesser known school would always be in a disadvantage compared to say graduates of good US universities, as I would not necessarily know the letter-writers and how to evaluate them accordingly (being top of class is nice, but I need to know about the class). This is relevant even for people from the best schools in China, as not all the faculty of even the schools I have mentioned are widely known in the west.
Considering the fact that there are ~100 places in the top-6 grad programs per year, and you are competing with the best graduates of US universities, Canadian universities and some European, Israeli and Indian universities, you see that it is EXTREMELY HARD to get into those programs, even for really good students, and a lot of this rides on your letters of recommendation and pure luck.
I would say that your best strategy would be to transfer to the better school as soon as possible (and obviously don't do the 5 year program, maybe only the first 3 years for BSc and then move to the better school for the MSc), as this will raise your chances, as you will be able to write the name of the better school on your CV later with the MSc degree. Moreover, doing grad courses (and hopefully some research) there will get you recommendations from professors at the better school, so most likely they are better known.
Anyhow, especially with your remark about the master graduates, I suspect the graduate program is not very good/not perceived like one in the West, so it is best to move to other places even for the masters as soon as possible.
If you want even more chances, pull out your cheque book and go to Princeton (I personally believe that you will get better mathematical education for BSc in Tsinghua, but it will be hard to deny a good letter from a Princeton professor, simply the way it is).
I don't think that continuing asking this question with epsilon-changes any week will help you or be productive.
A lot of good mathematicians got their education from not top-6 schools.
There are many upsides for going to top-6 schools, but also some downsides.
Again, if you are not happy where you are, you should move ASAP, and definitely not stay there for 5 years, as this will lead to depression and for you being counter-productive.