When it comes to EURO, in 40-50 years prior Southgate all they had was one semifinal, and when it comes to WC one 4th place, so its not like they were super successful. And not even when they had better team than they have now. Btw not defending Southgate, just telling you how possibly people that are deciding might look at the situation.
While back to back finals in the Euro is definitely progressive for England as a national team over the past half a century, we can't argue that this side have like the best caliber of players in every position and what they've achieved shouldn't be considered as over achievement because of the type of players they have at their disposal. Players like Palmer, Eze, Watkins and Toney wouldn't not be sitting on the bench for majority of the nation side that played in this year's tournament.
My biggest observation and opinion of this team is that the coach really needs to find the best way to integrate Jude Bellingham into his lineup, because if you ask me Foden had an awful tournament, and Saka could have it better too, but in my opinion the Bellingham-Foden forced partnership literally inhibited Foden's performance, and as such Foden would have been better coming of the bench, Bellingham was played almost as a No 9. in Real Madrid many times and it worked well for them, while Bellingham presence is a very evident within the opponents box, he's not as creative as you'd see with players that keeps possession to create chance, that's why Pedri and Dani Olmo worked perfectly for Spain, because there is a Fabian Ruiz and Rodri, in the midfield both. England found it really hard to create chances off possession, and that's because because Bellingham doesn't play a possession based game, while that's where their best link up could have been Rice and Mainoo --> Bellingham, then Bellingham ---> Foden and Saka.
I'd even say if they must start Bellingham and Foden then England best lineup should imo be a 4-2-3-1 with Pickford; Defenders; Mainoo, Rice; Foden, Palmer, Saka; Bellingham.