Does your TV not support HDMI-CEC? When enabled devices like Chromecast will send a command to the TV when someone initiates a cast to turn on the TV and switch to it’s input automatically. No need for OH to be involved at all.
Not all TVs support this and it isn’t always enabled by default. And there might be cases where you wouldn’t want to enable it. For example, I have a Rule that sends my bedroom Roku to the home screen at midnight to stop all nigh streaming if we happened to fall asleep with the TV on. We set a timer on the TV to turn off. But when the command to go home on the Roku was received it turned the TV back on.