The last World Cup and Euro winning teams have managers who were their national team’s U21/20 team managers before. Even Southgate, for all his faults, lead to our best results in tournaments came from the U21s.
Then consider winning managers like Löw and Santos had experience in the international setup either internally or with other nations.
The managers with success at club football, like Deschamps and Mancini, wouldn’t be considered ‘elite’ in the club game when they become international managers.
Final point, Germany have tried hiring elite managers like Nagelsmann and Flick, with both of them failing to get deep into tournaments. Nagelsmann gets the BOD for losing to Spain but as a host nation…
Hiring managers with international experience seems to be a better route to success than elite club managers new to the international scene.
The FA probably panicked when Carsley lost to Greece to hire Tuchel. Carsley also gave some odd responses in the media so perhaps the senior role was perhaps too soon.