Find all object type children of a bone

If I understand this correctly you want the children of an armatures bones.
In that case the children are still the armature objects children. The bones are specified as obj.parent_bone given that obj.parent_type == “BONE”.

Edit: Changed obj.parent_type (mistake)
Here is a long version of how to handle it:

import bpy


List = []

ARM = bpy.data.objects["Armature"]
    
for BONE in ARM.data.bones:       # search through bones
    
    Bonechildren = [] 
    
    for CHILD in ARM.children:    # check if there are children of the armature that have the bone as parent_bone 
                       
        if CHILD.parent_type == "BONE" and CHILD.parent_bone == BONE.name:      # check for the name string, not bone data 
                              
            Bonechildren.append(CHILD)   
                                                 
    List.append([BONE, Bonechildren])
                       
print(List)  # [ [bone1, bone1_children] , [bone2, bone2_children] ... ] 
for x in List: print(x)