I believe there is a Google TV set-top box from Logitech that does exactly what you are saying, where it will connect to your cable box or whatever and function as a DVR. The only problem with this is that it will use an IR blaster to control your cable box. It's a horrible setup which requires you to program things and set it up correctly. Plus it still doesn't get rid of your cable box. Even still you basically are still dealing with delays while you wait for the IR blaster to change the channel for you.
On the other hand I think DirecTV is working on a Google TV box as well, which will function directly as a DVR and skip the whole IR blaster deal. The only problem with this is that not everyone wants DirecTV. Plus nobody knows how limited it will be, if it will have all the same features as other Google TV boxes. They could choose to not fully support apps for example.
What's to stop other TV providers from making their own set-top boxes with Google TV and DVR functionality? Really it's mainly greed, power, and fear of the internet. A lot of the big cable providers (Comcast as an example) own quite a big portion of TV networks and channels, and so they have direct control over where those channels go and how they are used. If some provider wants to use those channels, they have to agree to the terms of network provider.
This is a huge complex issue that basically ends with the companies not wanting their content to be associated with the same box that lets you stream shows from Amazon, Hulu, etc. The stuff that is a direct competitor to advertising revenues from live TV channels.
This is why Apple is taking a different approach, they are going to the content providers and striking deals to offer TV rentals. This makes the providers happier because they see it as a new revenue stream instead of a replacement for the old one. Most providers are still very afraid of the internet though and don't really understand the potential (remember the music industry when iTunes first started?), and thats why Apple hasn't gotten more on-board (yet).
I really hope Apple's model works out honestly, because internet delivery is so stupidly obviously the future