Not really. Put him in charge of a midtable premier league club and he'd be sacked within a few few months/year. Deschamps and Scolari are hardly likely to get a call from Real Madrid anytime soon and they're World Cup winning managers. I wouldn't put it past the Glazers to consider him if things don't go well with Ole though
He's an ok manager, his man management seem excellent, and he has done a good job to get England where they are; he's also got an unusually fantastic group of players to pick and had some brilliant luck with routes to the finals, all the while playing a very conservative style of football. Good on him if he wins it but tactical genius (ITV commentators) and all the fawning is stretching it a bit. For years people have been having a go at England for lacking an identity in football that's expressive or unique and I don't think Southgate has changed that.
Also remember him being a right knobhead about Ronaldo years back