Yup, I am coming round to the opinion that Southgate is a natural and obvious successor to GP. I don't expect it to be a popular choice, I'm not sure how I'd feel about it myself.
But, the case for the opinion, if not necessarily the case for Southgate, is something like this : They both produce pretty similar styles of play and formations, three centre halves, wing backs, build from the back, defensively solid, all rather standard stuff perhaps but there would be complete continuity of style. Southgates strength with England seems to be his empathy with the players, building that team togetherness, emotional intelligence type stuff, don't get too high with the highs or too low with the lows, accept you have to suffer at times. Its all very Potter. He's said, or at least implied, that he wants to go back to club management sooner than later. If GP leaves us around 10th in the table we'd be about the right stature of club for him - higher than a relegation battle, but not quite at challenging for Europe stage, but with the potential clearly there. Dan Ashworth gave him the England managers job and he'd be leading the choice of new manager. And they both have a similar line in natty beards.
While the Dan Ashworth link makes it plausible and the team psychology aspects are similar (and important IMO), I think they have quite different attributes as coaches.
Potter is tactically adaptable for different opponents and makes in-game changes - Southgate pretty much sticks rigidly to one plan.
Potter limits opponents' attacking threat by coaching his side to retain possession and win it back quickly. Southgate sticks the handbrake on with 2 defensive midfielders.
It's not a fair comparison, given the differences in managing a club or international side, but Potter coaches players to make them better - Southgate just gets to pick the best (English) players and chuck them in his system.
Southgate wouldn't want it anyway - he'd have to pick Dunk!