You’re conflating something here. The statement only refers to “what is true”, not your situation; each pronoun refers only to “what is true”
In that case saying “Owning up to the truth doesn’t make the truth any worse” is correct, but doesn’t settle the issue at hand as much as people tend to think it does. We don’t just care about whether someone owning up to the truth makes the truth itself worse, which it obviously doesn’t. We also care about whether it makes their or other people’s situation worse, which it sometimes does.
In that case saying “Owning up to the truth doesn’t make the truth any worse” is correct, but doesn’t settle the issue at hand as much as people tend to think it does. We don’t just care about whether someone owning up to the truth makes the truth itself worse, which it obviously doesn’t. We also care about whether it makes their or other people’s situation worse, which it sometimes does.