Why a streaming stick has no input to change
A Roku Streaming Stick, an Express or an Ultra is a source, not a display. It produces one video signal and sends it down one HDMI cable into one port on the television. There is no second source for it to switch between, so the concept of changing input simply does not exist on those devices.
This causes real confusion because "my Roku" means two entirely different products. If yours is a stick and you want to reach the games console, the television has to make that change, and the Roku is genuinely unable to help. In that case the panel buttons on the TV or its own remote are the only routes, and the app is not the answer.