Bronze axes were just as good at chopping down trees as iron ones. The problem with bronze is that it was expensive. Iron was cheaper and more plentiful, enabling a lot more axes for a lot more people. Of course, in the Neolithic, they didn't have bronze axes either. In the Neolithic, chopping down trees would have been extremely time consuming and labour intensive. I think that one of the chieftain's roles would be to ensure that trees were cut down and land cleared to allow his people to expand. It would be a tangible and very visible indicator of his success as a leader.
Anyway, as the neolithic people had no metal tools, it's a wonder that they could clear trees. Even with modern high-quality tools it''s a heck of a lot of physical work to do. Did they take a large fallen log and use that as a kind of lever to uproot them? Maybe they had a practice of pulling branches off trees for firewood, thus slowly killing them.
But with their very low population, did they really have to clear land? Could they not just move on to the next valley? I would think that trampling livestock could be a major force in clearing land, or at least keeping existing pasture land clear.
Is there any real evidence that they did clear forests? I guess such evidence would consist of carbon-dated tree pollen samples showing the presence of trees before humans arrived, and none after they came.