Individuals can be oppressed too you know, not just groups.

And either way, the language "white people should be able to do ____" Implies, through common sense, that someone or something is stopping them from doing said thing.

Yes, it's pointing out hypocrisy (however imaginary that hypocrisy is), but that doesn't mean that someone else isn't stopping you from doing something you think you should be able to do.

For instance "women should be able to vote too" Is pointing out the hypocrisy between men and women (in the past), but that doesn't mean women were not also being oppressed. Obviously they were, they were being excluded from voting. In this case OP is being excluded from singing a song (in his imagination).

So yeah, it's implied that someone is stopping him from singing the song ( they aren't), so again.. The definition is "the state of being subject to unjust treatment or control". And Someone else or something is unjustly controlling what he can or can not sing. It's really not that complicated.