I would go for the simple idea:
Grassroot movements have to start bottom up. Leadership is 'nice', usually no one says thank you, everyone complains, you have meetings, discussions, managment complains ... welcome to leadership everything you do is wrong. Like being a parent... I heard.
You have to do it because you want it, not because an external source tells you to. And by staying consistent and motivated slowly there will be change. It will take time and a lot of work.
as Lazar Pavlovic pointed out the main thing is that you get your team on track and that you guys agree on something. only if you agree and talk about it there can be change. Also it's very important to do things outside of work even better cross department, but I don't know how big your company is and how the hierarchy works.
That's all I can contribute. If you want to lead you need to lead and this means also not giving up but constantly moving forward and with that you can inspire people.
Although it's very natural to get frustrated.
Btw the reward is usually 2-3 years later when someone for example skipps company and tells you what he learned to appreciate about you after seing the contrast of missing it.