I had heard people say this about Reagan. He took accountability for things that were arguably out of his control, and people later didn't blame him for things that were.
To me the lesson is accepting blame for a problem doesn't just take "Whose fault is this?" off of the table, but it increases the trust in finding a solution.
Yes, the act of making an apology will be perceived as implying that the speaker is more than willing to help improve the situation, regardless of how we got there, thus what you said.
By "taking accountability" you mean he said he was taking accountability without actually suffering any consequences. That's not taking accountability, that's lying about it. Same with Harry Truman "the buck stops here" -- OK, which of the many terrible things that have happened under your watch are you going to quit over?
To me the lesson is accepting blame for a problem doesn't just take "Whose fault is this?" off of the table, but it increases the trust in finding a solution.