Content that could potentially upset a viewer with no real value (e.g. gore, violence, on-screen sexual scenes, bad language) should only be shown at certain times of day on music channels, and should come with a warning. Otherwise they should be censored. It's important to consider how the content could affect someone sensitive to those kinds of things, and they should be given a chance to switch channels before they see or hear something triggering or disturbing.

It does depend on what's being labelled as "controversial" though. For example, homosexuality is often seen as controversial in media, while at the same time it's perfectly normal and can't mentally harm people like the above examples. In those cases, it's probably best to just show it without making any fuss, thus encouraging the viewers to accept it without thinking twice.

I just think people need to actually think about why they are or are not censoring something. Is the content harmful to some? If shown, should it come with warnings? Or will those warnings sometimes make matters worse?

I hate censorship of all kinds. Especially on music.

These are some examples of songs and music videos considered controversial.

whatculture.com/music/20-most-controversi­al-­son­gs-­of-­all­-ti­me.­php­
www.mandatory.com/.../the-most-controvers­ial­-so­ngs­-of­-al­l-t­ime­/
www.youtube.com/watch?v=mqe3P03Qthc
www.youtube.com/watch?v=kH25QOo6L9c