Potter has a history of developing teams playing with a decent playing style, passing out from the keeper etc.
Southgate had a history of doing fuck all with Middlesbrough.
That said, with the time allowed together at international level, it's difficult to get a team to learn a form of positional play considering they all come from teams who play different ways. There'd be enough players used to it, but if(when) Stones is injured the whole concept would fall apart to some extent.
Klopp's brand of football would be quicker and easier for players to pick up, so I'm sure he could be good.
However, he'd also find that the media is very much different for England than it is for the red cult, so he may not fancy the job.