Do you use a receiver to bridge between your TV and your devices (if you have a surround sound system, for example)?

Often receivers will have overlapping Input settings for both HDMI and AV cables. Both inputs on the back of the receiver have the same label (game, video, DVD, etc.) so you can use either cable type. In my experience, if both cable types are plugged in HDMI takes priority if it is giving a signal.

To make a long story short it could be that your GameCube and your Switch are plugged in under the same input of your receiver, so when you take the switch out (and the HDMI is no longer sending a signal) the GameCube replaces the input.