I wouldn't hold my breath waiting for a solution. It isn't in the cable/satellite companies interest to make it easy for you to make your own PVR or distribute all streams to all your TVs from a single box. They want you to buy their own PVR system (if available) or multiple boxes along with the associated limitations and fees.

The analog TV system works the way you want because there is a tuner inside every TV and there is generally no encryption or protection systems.

The US has the previously mentioned CableCARD system which would mostly do what you want but thats making the set-top box tiny basically. You still need 1 CableCARD per device.

Making a single set-top box that was able to tune into every signal available + decrypt + distribute and optionally do PVR duties would be incredibly complicated and expensive.