The song to me feels more to be about how the intergenerational cycle of abuse isn't a 1-to-1 transfer of trauma rather than being about any refusal to continue the cycle. Sue's trauma is about chronic mocking/bullying resulting in an insecure need to prove his masculinity via violence. His father's struggle isn't specifically elaborated on but it's clearly present and ultimately comes down to a similar relationship with violence/masculinity. His abandonment and rash naming decision caused his son a lot of struggle in his life and his father's reasoning for doing that was because of his own life's trauma.

The conclusion to the song was Sue coming to understand and appreciate his father's decision even if the trauma Sue experienced prevents him from repeating the decision. Which underscores the way intergenerational trauma and violence often isn't a straight line. Sue didn't decide to become a great father who would keep his children free from the need to constantly prove themselves physically. Sue merely decided not to name his son Sue.

Only I some versions. In some the singer decided to name their child sue as well, perpetuating the trauma

You should listen to the second song, then. Might change your perspective.

Full disclosure: both were written by Shel Silversein, but Cash refused to perform the latter. This is pre-Where The Sidewalk Ends, mind you. This is The Great Smoke-Off era Silverstein.

edit: Here's Shel Silverstein performing "Father of A Boy Named Sue"

You should check out the song "Patches" If that's what you're interested in.