Back then they had Scholes. After that they relied on deep defensive midfielders like Lampard and Gerard, which played far too deep in the back to make an effective attacks, England was very unproductive at scoring goals. Rooney was too traditional as a striker to play as a false 10.
Today, they have Kieran Trippeir, I think he could be the modern day Scholes.
Dele Ali was physically limited in this tournament, he was great in England's first game, but then after contacting that thigh injury, his pace was greatly affected.
Raheem Sterling, despite his matured age, still lacks in vision, which is why despite his dangeorus and imposing presence, his contribution to goal is minimal.
Ruben Loftus-Cheek is another surprisingly good player. He is big, solid, can dribble, and have all the unique passing skills necessary to play English attacking style. He could be a playmaker.
For English defense, I agree with you, they are full of holes. They often play too much risk with their dangerously close passing when the enemy is still pressing. But then, they are always like that since the day I first watch football with Sol Campbell, Paul Ince, Southgate himself, and Neville days.