They don't always. In fact in South Africa the alien, invasive tree (more properly speaking, giant herbaceous plant, or forb) Phytolacca dioica, is regarded as a pest in forestry in certain regions. Certain fairly large pigeons, such as the beautiful Rameron pigeon, will gorge themselves on its berries, and perch on the fragile leader shoot of young pine trees. These sometimes break, not killing the tree, but rendering it useless for purposes of forestry.

The silly pigeon is not silly enough to fall, but flies off intact, ready for the next performance.

Actually, in my experience pigeons are not at all really silly, so I suspect that not many of them make a habit of perching on the leader shoot if that is anything like a routine result.

But most twigs on most branches are strong enough to support most birds, so it is not something that one expects to happen often. However some bird pests, such as the African red-billed finch or Quelea, have been known to perch on branches in such masses as to cause them to break. I have never seen that myself, but see no reason to doubt that it might happen.