You'd have to be very naive to believe that money wasn't a major factor, if not the deciding factor, but other things will have come into mind.
From an outsider point of view, Wenger seems to run the club in a very methodical and calculated way, which obviously has it's pro's and con's. I don't think he is a great man manager, and he certainly doesn't come across as an authoritative figure. I know that SAF makes sure every new signing understands the history and culture of United, and that they respect the traditions and older statesmen. There was quite a large number of foreigners coming into Arsenal at the time, and perhaps Cole needed the feeling of belonging as well as the discipline and respect that it seems Wenger doesn't always command?
I'm sure Wenger and Arsenal could have kept Cole if they really wanted to, but the manager decided to stick to his own principles, for better or worse. I really don't like the lack of loyalty that Cole showed the club, but I'm not sure that I would have felt differently had I been in his shoes. I'm sure he probably had more friends at Chelsea than Arsenal at that time. The situation with Nasri is totally different to the alienation that Cole may have been feeling, but the disloyalty is still the same, and if I were an Arsenal fan I'd feel the same disappointment.
At the end of the day, SAF opted to extend our wage cap to meet Rooney's demands because he is that important to our football club. Was Cole that important? Is Nasri also as important? With Wenger's youth policy, there is always a youngster ready to step into their shoes.
Apologies for the long(ish) post, must have Curtis syndrome or something