I've done a full 180 regarding Southgate in just a few months. I can see him staying for a long time.
Having not thought much of Southgate as manager of Boro, I respect the fellow's achievements with the national team to date. He gives the impression of managing as he steered himself through his playing career- aware of his limitations, but working hard and thinking carefully how to improve. He's also the first England manager in a long time who appears to be working in the present day game, rather than trying to hammer it into a shape that suits his failure to evolve.
I also think his low-key diligence meshes nicely with a team happily less burdened by rampant egos than its predecessors. It must help that while he himself must have an ego - no man can survive as long as he has in the professional game without one- he conceals it well. In some ways, the team's progress has been almost Allardycian in its dependency on not conceding a lot, while scoring a high proportion of goals from set pieces - but, crucially, we've been spared Allardyce's chippy, have-his-cake-and-eat-it self-promotion.
Southgate had the advantage of low expectations entering the tournament, and the order of England's fixtures in the group stage was fortunate in giving him leeway for a positive vibe to build. But his efforts will stand or fall on who England eventually lose to in this tournament, as well as the manner of the defeat. Sterling, for instance, appears to be on the brink of becoming this generation's John Barnes, and his manager may yet take a kicking for sticking by him.
I can understand why some fans are frustrated with the wing-back formation, but I consider it the lesser of two evils. As long as the English work on the basis that playing central midfield properly is something best left to foreigners, they'll always struggle to field a 4-2-3-1 formation that isn't a monstrosity.