The phrase "promote the message of" is actually 2 parts.
To promote: support or actively encourage (From Google definition)
The message of: the main idea that an artist, writer, speaker, or group is trying to communicate. (From https://dictionary.cambridge.org/dictionary/english/message)
So putting them together means you are supporting or encouraging the main idea of the subject. So to promote the message of health and safety would be to encourage people to follow or learn about the main points regarding health and safety.
To promote the message of Chinese culture would be a perfectly natural sentence. It would mean to help encourage, support or spread the main ideas or beliefs of Chinese culture.
Promoting the idea of Chinese culture is also a natural sentence but there is a subtle difference. Promoting an idea generally means to make people think better of something instead of spreading the idea to them.
For example, a Christian might promote the message of Christianity if they wanted more people to follow Christianity and they might promote the idea of Christianity if they wanted people to be more tolerant of Christianity without actively following it themselves. These often go hand in hand though so the phrases can become somewhat interchangeable in some situations.
Depending on your specific aims, either of your sentences could be appropriate.