There are two sides to that. I was representing a German union institution inside our org for one simple reason. The employer had gone out of his way to ignore what peoples responsibilities were and showered them with work that wasn't there's and then tried to punish them for pushing back. Usually the reason people respond with this is not my responsibility is because they have been burned by doing things that wasn't theirs. It's a cultural issue and the solution which you describe is a crappy generic solution to a cultural problem.
That said the very same union would frequently respond with that's not our responsibility when I pointed them to take up actual legal issues that they were responsible for.
A lot of the old union representatives were there for decades with no desire to change anything and enjoying the good relationship they had built with the leadership.
At the end of the day any human organization that grows too big seems to end up in its own lazy comfort zone, whether it's the employer side or the employee representing union side.
The thing about forming your own union is that major unions also have big lobbying institutions saying that their union is the only thing with real power, while the other side has their own lobbying groups pressuring workers into believing that all unions are bad.