It does NOT mean "He agrees to settle any financial claims against himself or his employer, for damages caused by negligence".
Could have said "I blame only myself for this mistake, nothing else needs to change".
This will keep happening until the system where this can happen runs out of willing scapegoats.
What does “I take full responsibility” mean if not (amongst other things) “I accept it is my duty to pay reparations”? As you yourself rephrase it “he could have said ‘I blame only myself for this mistake, nothing else has to change’”. That would have been sufficient and beyond necessary. “I accept full responsibility” is corporate-speak empty hyperbole, and it has to end. Is he any more responsible than those who introduced the hole in the first place? No. Is he responsible for not adhering to agreements he was not privy to? No. He cannot “take full responsibility” because that is an utterly unreasonable burden to place upon him, a lone well-meaning actor who unbeknownst to him took a misstep. We need to discount this corporate-speak linguistic hyperbole.