I think that's the kind of manager they need - as much of a psychiatrist as a tactician. Chris Coleman is hardly a tactical genius, but he got his best players playing with freedom and confidence, and his worst players busting a gut for the team. Pretty much the same goes for both Ireland teams and Iceland.
It might even be worth, as an experiment, an England manager eschewing the traditional view that the team should be some kind of stellar showcase for the most sublime talents in the land, and focus instead on picking personalities who'll function best as a unit. After all, Ramsey didn't fill his 1966 team, or even squad, with the very best players. He had a spine of top class talent, in Banks, Moore, Charlton and Greaves'Hunt, but most of the others were reliably functional. If Allardyce did get the job I'd assume that would be his modus operandi. And, while that might well depress some fans, and certainly me, it would probably produce a team that could do as well as Wales.
If you go the other way and stick with the increasingly arrogant view that an England team has not just got to win but win 'in the right way,' then you'd surely need to find a Venables type who could somehow execute the trick of making enough of the players believe that they're as good as their reputations suggest, and buy into a clear and coherent tactical plan. And I don't see an English manager who can do that.