I found no evidence for this, arguments seems to boil down to an "appeal to nature". Many variations of bodyweight exercices are far from healthy for tendons (single arm push ups or single arm pull ups for example), even basic push ups are harder on the shoulders than dumbbell bench press.
If you keep the rep range above 15-20 the risk of injury is pretty minimal with weights.
With that said, bodyweight exercises can be a good alternative if you don't have access to a gym, or if you like to train outside on a playground. But be careful with some exercises because bodyweight != healthy.
