It doesn’t need to be load-bearing because the actual forces aren’t that high. Fifty pound drywall anchors exist, after about fifty pounds the child will likely be old enough to not climb, and a fifty pound child’s angular force pulling out on the wall is going to be less than fifty pounds since most of the weight is going into the floor.

Certainly anchoring to the stud would be better, but drywall anchors are better than nothing and should probably be sufficient in most cases.