1. Environmental -- For "our children" as you call it. This assumes that countries have gov'ts that give a shit and are not totally controlled by the industry (essentially the mega shareholders)
2. Health -- Plant-based (especially whole plant food) diet is much, much healthier than the a diet that contains several serving of meat/dairy/eggs per week (many have that in a day). This is ALSO an argument for the children, as they will learn from us. (We could not tell kids to not smoke, we have to stop ourselves too!)
3. Ethical -- Whole categories of abuse towards fellow earthlings (poor, women, slaves, lower-caste, captured enemies, etc.) were generally accepted as permissible, yet are now near unanimously condemned. It is not unfathomable that animals --that are currently still being permissibly abused for being of a different species-- may get added to that list.
Also the ethical reason will have a huge impact on "our children". As we have seen with the other steps we made in awareness, it is totally worth making these steps.
Peace.
There's a huge influx of people in the vegan/plant-based communities from very health conscious people. Many of them believe that science is showing that it allows them to live healthier/happier/longer/stronger.
Is that not a good answer to "What's in it for me?" :)
But you're advocating the one blow will be from the "personal health" angle. With little time on my hands I usually also go for this angle, simply because I believe that all the people who'd want to be vegan/vegetarian "for the animals" would be so by now. The health angle still has a kind of novelty/news aspect to it (while being more personal and urgent then the environment angle).
Long to short, factor farming generates a lot of ground level pollution. Furthermore, those animals also generate a lot of greenhouse gases. Frame the problem in terms of human suffering and more people are likely to pay attention.
Years ago there was a NY Times Sunday magazine feature article on factor pig farming. It was enlightening. Unfortunately, I've searched for it and can't seem to find it. If you can find, do read it.
p.s. Growing crops to be fed to farm animals is also very inefficient. More people would have more food if we all ate less animal protein.