Would have thought this was feasible and might be a popular option.
One of the problems with the Tivo is that you can sometimes rewind a channel, but not always - it depends what the tuner has been doing before you started watching. So sometimes if something grabs your interest you can get the whole show even if you started watching partway through, other times you can only record from where you recorded from.
How hard would it be to set up the system so that if the Tivo box could not record from the start of the show, it would prompt with a "Do you want to watch this show from the start?" and then if the answer is yes, have a little chat with the servers and download the programme from the start, creating the recording on the fly - whether it merges with the current TV stream or simply streams the whole program in the background, your techies could decide.
You don't always have shows on catch-up or they take too long to filter through. It would be a nice little feature to trump the opposition.