That would be a terrible move. It gets far more exposure as a standalone event than it ever would if it was taking place at the same time as the main Olympic competition which has over 20 sports taking place. I imagine it would also be a logistical nightmare scheduling both events simultaneously.
I've warmed to Corden over the years. I think he's one of life's good guys and he is a versatile performer with a sharp mind.
The cocky stuff I think is more his stage persona, rather than his actual personality.
Great poetic script by Eddie Butler, who came up with a brilliant piece of oratory at the end of London 2012 when they showed about 45 minutes' montage of the best bits from the Games.