I don't think it would be easy to use a full frame sensor for one second.
Aside from the fact that the sensor cost increases exponentially with size, there would be more data to process and store which would increase the memory and power requirements. This all adds to the size, cost and weight of the camera and people complaining it about the lack of FF would instead complain about the cost, if that isn't high enough already.
And that's before they try to address the image fall-off problems Paul refers to.