Dick, sector sizes are based around workload - providing an approach service is more workload intensive than simply providing traffic so to cater for this a sector will need to be smaller. Also providing an approach service on a 500NM wide screen is hardly ideal and do you really want the controller fixated on the approach service (out of necessity) in one corner of his screen while ignoring the other stuff is going on elsewhere? Would you hand fly an ILS approach in ****ty weather and try to play a game on your phone at the same time?
Maybe, just maybe they can do it in the US because their sectors are the right size to handle it. You're not wanting to copy the whole system - you're wanting to shoehorn it in regardless of whether it will actually fit into the current infrastructure.
Maybe you need to get that into your fixed mind......
Show me that it can be done with current resources. Show me where you are going to get the extra controllers to allow for training for the approach rating - training requires time away from the console so someone needs to be available to fill the seat. Where are we going to source all the instructors, the sim time.
I know you won't because you've been asked this countless times in the past and never answer.