Does the TV stay on when the xbox one is not hooked up? Seems to be the case per your "chromecast" comment.
If so, the problem may not be the TV.
Consider that the gaming "load" on the Insignia TV is causing some overheating and as the TV gets increasingly hotter something eventually shuts it down.
Check the TV's ventilation and ensure that full air flow is available.
Try the xbox on another TV and determine if the problem follows the xbox or not.
Or a different xbox on the insignia TV if at all possible.
Objective is to eliminate or narrow down possible sources and then the specific cause.