I have three villagers pushing the statue and it's going very slow. They will not get it done before they die. I'm worried about getting enough people trained to push the statue later. Does there always need to be a master builder present or just to start the task? How will I get enough trained builders when these guys pass away? Everything's built.

Do I have to remove the rubble, and the leaves, before I can start pushing the statue?

The only villagers required for the statue are the chief to direct and at least trainee builders to push (once the plans have been drawn and the work has begun).

The rubble and the leaves are great places to train builders, but they are not required for the statue.

Well, if you want to train villagers in the building skill an you have everything built, just drag them onto the existing huts. It will often display that they are "fixing hut". If they succeed they gain some building skill.