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I love that analogy. No, you're not going to personally save the world by reducing your carbon footprint. But you know what you are going to be? One leaf on a tree in a forest, making that little bit more oxygen that helps collectively make the world a better place.
And that's worth doing. Especially if you can encourage other people to be leaves too.
As anecdotal evidence, I live in a Brazilian region that produces a lot of cattle. About 10 years ago, beef production went down due to economic reasons, and this was noticeable in land cover studies, The mere reduction in cattle numbers, grazing intensity and pasture management was enough for the forest vegetation to advance and take back some border areas.
So, this is very true, and not just "vegan propaganda". Reducing demand for meat can have a significant environmental impact!