Seriously folks, this is how little sense it makes not just to keep him on but that he ever got the job in the first place:
Southgate was appointed Middlesboro manger despite not having gained the required professional coaching qualifications, or even taken the exam. Boro successfully argued that his international commitments (as a player) meant that he had not had the time to do the course.
In his first season his team lost away to all 3 promoted teams, and didn't win an away game until January, against struggling Charlton who were relegated that season. His first 4 signings were defenders. The club spent some time in the relegation zone but finished 12th, and the following season finished 13th. The season after that they were relegated. Southgate kept his job - just - but was sacked the next season after failing to get the club promoted back to the PL.
He then spent 4 years out of the game completely, before being appointed manager of the England U-21 team, from where he eventually got the full England manager's job after Big Sam was set up by the press and lost his job as a result.
On what basis does the FA figure that this guy was ever the best man to take the England manager's job? One short and unsuccessful stint as a club manager then 4 years out of the game? You couldn't make it up.