In the PL (playing) era, Paul Ince is probably the most successful manager of the former England players, with a L2 promotion.
That tells you everything!
Keegan is likely to be the next best, by honours.
As an aside, I just looked at Southgate’s wiki for his list of managerial achievements. He doesn’t even have a section! However, not too shabby as a player. Although never played for a ‘top side’, he has been captain at every club he has been at and won promotion to the PL (Palace), League Cup (Villa), League Cup (Boro) and a UEFA Cup finalist (Boro).