This is absolutely right.
Southgate has got the England team to overachieve - no English player is the most important player in a Premier League team, they are all either the supporting cast at Man City, Liverpoolt etc., or the stars of middling teams like Palace. Southgate has managed to get this squad of not-greats to play as a team, something not managed by many of his recent predecessors and as a result we've been genuinely comeptitive in the last three major championships. I don't think I can name three other successive championships that this is true for.
International management is different to club management, Southgate did his time with the u-21s and I consider that far more relevant than anything he did at club level. If he does go over the summer, the FA should consider the same route for his succssor.