They'd have a decent shot of winning the tournament even if their talent lagged significantly behind the top 3. A lot of players suited to play in a fast-paced pressing battle.
I don't think I'd like a Rooney-Alli-Wilshere midfield on balance. Rooney as a 6 seems like a far bigger risk than even the playmakers who sometimes draw criticism for their poor defensive play (e.g. Kroos and Alonso). Dier is a solid 6, but he's not going to control the game either. I think at this point I'd just go for Dier-Rooney-Alli like they played today. It didn't necessarily look pretty though. Connection between Rooney and Alli was pretty poor. I guess Rooney is more comfortable making passes from the left half space, but that English left side (Rose+Sterling) is athletic but didn't really generate danger. The group of players they had on the field is ok (Sterling and Smalling are the only two I'd probably change) but it doesn't quite fit together in possession yet. Good pressing though.
I like Dier, but Rooney and Alli would need to improve their connection massively for him to work in a midfield trio. Personally I'd play my beloved 4-3-1-2 with Dier behind Alli & Wilshere, and Rooney in a free floating playmaker role between them and whichever combination of Kane/Vardy/Sturridge shows the best chemistry. But Woy won't change things up that drastically during a tourney.
If tied to the 4-3-3, then I just don't think it'll work with Dier at this point. A Rooney/Alli/Wilshere duo has the talent to make it work, but they don't seem to have the mutual understanding to do it. That's why I'd want all three to play together to give them the skill to create consistently & fluidly and to overcome their issues creatively.
That trio's defensive frailties is why I'd use Lallana & Milner as narrow wingers; to help shield them & lead the press in front of them. With Walker & Rose's strengths lying in their pace & attacking capabilities, they'd have the attacking width to support that sort of front 6.
Kane was badly integrated into the match and didn't get good service.
I agree that the biggest issue was the poor connection between Rooney & Alli in midfield, which lead to the poor service, but Kane was very poor in his own right. I'd give a lot of thought to giving Sturridge or Vardy the start in the next match if Hodgson's not going to tweak the shape.