I just don't see how you can describe his role at Chelsea as a "bit part".
The key players, in terms of influence and contribution, for Chelsea in that period were Drogba, Fat Frank, Makelele, Robben (when fit), Terry, Carvalho, Cech and probably Essien.
It's then quite a big gap between them and the next set of players - Cole, Duff, Gudjohnsen, Malouda, Gallas etc.
Call it what you want - bit part/2nd tier/less influential. My point is about how it would have been nice to see what he could have become as a player had he been playing in a role and team more suited to bringing out the best in his skillset, rather than him not playing to his potential to fit around others.
As for the bold, in this era we're discussing, there really wasn't as far as England goes. It was the Big Four and a gigantic chasm below them. If he'd have moved to Everton, Spurs or Villa, we'd be discussing wasting talent on an enormous scale.
If, for some reason in this hypothetical scenario, he wasn't allowed to play outside of England, Newcastle could have been a good fit - Shearer, Bellamy, Speed, Milner, Robert, Woodgate, etc in the mid 00s - quite a decent team to build around someone like Cole to see if he really was the outrageous talent that he looked as a kid.
I look at another phenomenal talent - Riquelme - and see what he did at Villareal. Now that's an example of someone who wasn't good enough to build a huge team around, but build a small/mid size team around him and the world got to enjoy a few years of an absolutely wonderful player on top of his game and playing the football he was meant to play. In terms of natural talent I think Cole is up there with him. Neutrals around the world and hipsters still talk about Riquelme in hushed tones though - I don't think the same can be said for Cole.
to be honest I don't know what is better - to be the 8th best player in a side that wins a few trophies or being Riquelme at Villareal. I can see the appeal in both. As he was a Chelsea fan, and as Chelsea paid probably 4 times more than any other team would have, that probably swings it to the former. But in terms of legacy left on the game, it's probably the latter.