Why can't nations like USA, China, Japan take part in the Commonwealth games?

The short answer to this is:

Common Wealth Games is played among the 53 member states that where formerly under British Empire. Since nations like China, Japan, where never under British empire. They are not part of it.

More facts 1:

Commonwealth games are open for former British colonies only. 

China and Japan have never become a colony of Great Britain in the history.

As for US, it has been a superpower since several decades ago.

Taking part in the Commonwealth games would only remind US that it was ever colonized by Great Britain. So for the pride , US is also shamed to take part.

More facts 2:

The Commonwealth of Nations, commonly known as The Commonwealth (formerly the British Commonwealth),is an intergovernmental organization of 53 member states that were mostly territories of the former British Empire. The Commonwealth operates by intergovernmental consensus of the member states, organized through the Commonwealth Secretariat, and non-governmental organizations, organized through the Commonwealth Foundation.