One thing I would consider doing, if I ever start my own company or rise high enough up in someone else's company to implement this, is to give every single employee some sort of discretionary budget to spend on things that make the workplace better. The more experienced and trusted you are, the bigger of a budget you get. I think this would go a long way in fighting the learned helplessness dynamic.
20% is not specifically for "improving the organisation" of course. It's most famous on the outside for the new products it led to, like Gmail. But actually most 20% projects were small internal things intended to smooth the rough edges off a particular tool or process, or to improve the company in some way. If the particular bee in your bonnet was a type of bug that cropped up frequently in other people's software, making a linter to spot it and driving adoption through the organisation would be a good 20% project, for instance.
There may be awesome companies out there where anyone is empowered to just go fix some practice that's not productive, no matter where they are in the org chart, but those companies are few and far between. I'd love to see one.