The issue with Southgate is that the English national team has progressed beyond him. On the surface, the results (2018 WC - Semi Final, 2020 Euros - Runners up) are great. But in context, it's a bit disappointing.
As a neutral - throughout my life (I am 33 for reference), England has always been talented, but bit overhyped. Some wonderfully skilled individual players, but flawed teams that never really gelled or were a step below the top European and South American teams of the era.
That all changed around the 2018 World Cup. The current English team/generation of players is legitimately one of, if the most, talented and technically skilled sides in the world. We're just so used to England underperforming that the runs you've had in the past two tournaments have disguised that's what this side should be achieving with the players available.
Look how Luis Henrique had Spain humming the other day. England, on paper, is more talented than Spain. But under Southgate - they will never, ever play like that. Southgate has them playing uninspired and boring, and quite frankly, they're winning in spite of him.