Is what the father does, cleaning the places where his six-year-old son sits, who wipes the place of impurity from urine and does not perform ghusl, correct, or is there no need for that?

The fundamental principle is that children's clothes and bodies are pure until their impurity is established with certainty. If the child wipes away the impurity and removes its substance with a tissue or similar item, this is sufficient for purification (istinja'). What the questioner does by cleaning the child's sitting places and believing them to be impure is misplaced, and one should be wary of obsessive whispering (waswasa).