I like the sticker teaser.
Unfortunately, as you noticed, people on Earth are too stupid to even become aware of this choice. Considering this choice is obviously too advanced for humanity on Earth.
Colony of ants always acts in the direction of common good, sustainable existence in the presence of limited resources and their leadership is not elected based on false promises... Colony of ants does not live in debt to who-knows-whom because their society does not depend on money.
Humanity on Earth has adopted exactly the opposite direction: endless fighting (for nonsensical reasons), abuse of limited resources (we need ~4.5 Earths to continue what we do here), electing conmen "leaders" who create planetary embarrassment every time they say something and making all choices and activities dependent on borrowed money.
Your idea about educating people to recognise propaganda is good, but you cannot count on the mainstream education system to deliver it. I worked in the education system for several decades and left because the system has little to do with education as I see it. The current system is designed to produce perfect members of the flock and any attempt to criticize the system results in consequences reminiscent of Great Inquisition.
So, if you want to spread the idea of "nobody goes to wars" you need to do it using an alternative delivery of information .
My idea, back in 1999, was to write The Freedom of Choice book to define the Purpose of Existence and give people "filter" for all crap (not only for propaganda) based on The Purpose.
I really do not know what else to do. If you have a good idea how to inspire people to think, act on it and let me know about it.
Nature has a special method to teach stubborn and decadent people on planets like ours: suffering. When conscious evolution system becomes corrupt - most people perish in a magnificent cataclysm. We are not the first "civilization" on Earth to perish, but we can be the last...