Thanks for the feedback. Are you running app version 22.214.171.124? There was an update recently.
The live TV channel that was previously playing when you were last in the app will resume in the background when you open the app again, but you'll be on the home screen. Pressing the back button once will bring the top navigation menu into focus and you can navigate to Live TV, On Demand etc., pressing back a second time will exit you to the live player on the channel you're tuned to. Is this not what you're experiencing?
One suggestion I have is to enable HDMI-CEC in settings on both the Chromecast with Google TV and your TV. This way when you turn off your TV, the dongle will be powered off as well, and when you turn it back on, everything will power up as well. The Chromecast with Google TV does draw about 8 Watts of power, so you might as well power it down if your TV is off.
Thanks @KHR, updating the app seems to have fixed the issue.
I was on version 126.96.36.199. I already had HDMI-CEC enabled on both the TV and the dongle. The problem I had with the old version of the app was that when resuming it would go directly to the live player on the last channel and none of the buttons on the remote would respond at that point except for the home button. After the update it correctly goes to the Telus TV home screen on resume like you outlined.