@Robbie
If I remember right all the way back in 1966 for the World Cup, the England Team only had 6 weeks together before the Finals Began, so further proof Alf Ramsey was a gifted Coach. Part of Ramsey's success was the fact he had an idea of the England Team he was going to put out for all the Matches, Group stages plus beyond, and I believe if it wasn't for the injury to Jimmy Greaves, Geoff Hurst wouldn't have got his chance in the starting line up. In fact when Greaves was declared fit, Ramsey still stuck with Hurst and the Rest as they say is History.
Regarding your idea of a team selected from three teams, which teams would you choose? Because the problem is, most Premier League teams Nowadays are Dominated by Foreign Players and English Talent needs to move abroad to get games against Top Class Players. Kane Sancho and Bellingham play abroad, but for different teams, do you leave them out of the England Team? And can you imagine the Uproar in the Media if the England Head Coach did? Also if I remember correctly an England Manager picked an England team with either 5 or 7 Liverpool players in it, because England needed to Win an Important Qualifying Match to get to the World Cup Finals (during the 70's the Argentina tournament in 78 I think it was) but that failed, as the Three Lions either Lost or Drew the Match they needed to win.
Harking back to the time of when Walter Winterbottom was England Manager, he didn't pick the England team, that was done by a Selection Committee, could that even be the Case Today? The Committee select the Squad Members then it's up to the Head Coach to turn them into a Team? Could that be why the FA Only select Yes Men? And Managers who go against this protocol are either never chosen i.e Brian Clough or gotten Rid of i.e Big Sam because both Men wanted to Manage England Their Way?