Private language schools sell the image of English as much (if not more) than they sell actual English education. As a Chinese-looking person, you would likely make a poor salesman of the English image, and don't have a snowball's chance of getting hired. However, there are other routes you may be able to take to teach in China if you'd like.
There are four main types of English Teaching jobs in Southeast Asia:
- Private Language Schools
- Private International Schools
- Private Universities
- Government-run Organizations
Private Language Schools
These are by far the most common. There are many different types of these schools, from cram schools (juku in Japan, hagwon in Korea, buxiban in China), to conversational schools to everything in between. Generally speaking, "native" speakers are hired to teach English conversation.
These schools are 100% for-profit, and sell the image of English as much as they sell actual language training. Many use deceptive billing practices, complicated contracts, and high-pressure sales to make money without providing much in the way of service (see Nova, among others). Many are shady and do not provide their employees with the wages or benefits promised, and sometimes don't pay their employees at all. Point is, a lot of these places are fraud wrapped in pretty wrapping.
This works because many customers are just as interested in the feeling of English as they are in actually learning the language. As a result, a lot of these companies will make more money hiring a non-native English speaker who fits the image than they will with an extraordinary teacher who doesn't. And the image of English is Caucasian in Southeast Asia.
For these companies, no amount of changing your name will help. You are not the product they are looking for. You are not going to help them sucker students in and milk them for all they're worth. Sorry.
Note: There are plenty of good schools out there I'm sure, who actually focus on teaching English. They are also the minority. China < Korea < Japan when it comes to 'legitimacy' of English teaching jobs, and the major Japanese chains are going bankrupt and/or contracting because of a drop in demand. Not saying that Japan is better than Korea or China, just that from an English teaching perspective, the general consensus from people who have taught in multiple countries is that the working conditions follow that hierarchy
Private International Schools
Most private international schools are popular with expats and are located in more urban places. They are also usually swimming with applicants. There may be exceptions, but for the most part these schools will not care too much about your ethnicity as long as you are a qualified and experienced teacher of the subject(s) they are hiring for. Qualifications depend on the country and school.
Since you are talking about teaching English and sending out many applications, I am guessing this category doesn't apply to you (if you are a qualified teacher, however, I would suggest giving this a try).
You can also get hired by universities if you have the necessary qualifications (usually a Masters in Language Education or something similar). However, this path is also fraught with all sorts of danger. At least in Japan, foreign teachers are almost always put on contracts, and will not receive tenure at a Japanese university. While the pay is good in the meantime, the nature of Japanese employment law means that they are reluctant to hire someone they can't get rid of in the future, and due to the nature of university (students are only there a few years), there aren't so many consequences to rotating professors in and out to minimize long-term risk to the school.
This may be different in China and Korea, but my guess is that you aren't applying for university jobs.
The Japanese government hires foreigners to teach on the JET Programme each year. You are assigned as an assistant language teacher in a public school, but due to education laws are always going to be an assistant. There may be similar programs in Korea or China, but if so I have never heard of them. At any rate, at least for the JET Programme there doesn't seem to be a bias against people of Japanese descent (there are plenty of 2nd and 3rd generation Japanese, many of whom know the language). If there is a Chinese version, this may be a good bet to have a chance, but I wouldn't hold your breath.