I've had a play with profiles, as Ben says not available yet for Tivo, and I know the app is pin protected.
Even if you could turn off suggestions it would be an improvement.
My 8 year old is still having nightmares from when she was at a sleepover with her friend. She watched a trailer for Chucky and it freaked her out.
I do trust my children not too watch such content, my 12 year old is a scaredy cat too
, but there doesn't seem to be anything to stop suggestions coming up. for example if I watched Scream and the next day my 8 year old wanted to watch something she could be exposed to the image of Chucky as a suggestion.
I can't think of a way around that which is a pity.