Connecting home elliptical trainer to TV power
A different idea might be to use a "universal remote" programmed with your TV and then electrically actuate one of the buttons, perhaps using your relay idea above. That way, you could potentially hit different switches...for example, you could MUTE the TV instead of turning it off, or you could hit the SOURCE button, to change away from the cable signal, etc.  These ideas might be less harmful to the TV than power-switching it on and off repeatedly.