//Why were all the meetings, etc not conducted by a team of brexeteers who were the most competent and chosen from all parties. //
Because...
1)...all the Brexiteers fled at the first sign of responsibility.
2)... there is considerable disunity among Brexiteers (sometimes even within the same individual Brexiteer) about what Brexit should be.
As to the wider question: the referendum was to a very large extent a civil war within the Tory party. The whole reason the referendum came into existence was a promise made by Cameron because he was scared of losing UKIP votes. When the referendum actually came about, virtually all the major campaign leaders were either senior Tories or old Tories-in-exile like Farage. Outside of maybe Kate Hoey, the Labour party was pretty much absent.
So while the referendum wasn't a party-political issue in the traditional sense, it certainly was not divorced entirely from party politics.