I used to believe that in terms of convincing people to make changes in their diet which would help reduce animal suffering, using a variety of arguments was the best thing to do, as long as they were effective and got people to eat less meat/animal products in the short term.
I used a common shortcut notion of "the animals don't care why you aren't eating them", which, while not very accurate or nuanced, seemed to hold a certain amount of truth that made it valid.
However, after learning more on the topic and developing my views, I realized that the previous shortcut was not only inaccurate, it was probably actually a harmful idea to reinforce.
The inaccurate/simplistic aspect of the idea is fairly intuitive: animals are going to be killed, and reducing demand for their flesh just results in less being bred and brought into this world to suffer.
Thus, a non-existent individual cannot care one way or the other why they do not exist, and using an idea that lacks that nuance doesn't lead to an accurate concept of what vegetarianism or veganism result in.
However, how is it harmful? Well, if we acknowledge that vegetarianism doesn't "save" animals in a concrete sense, but rather reduces future suffering then we are in a better position to acknowledge the importance of the future more generally.
Thus, if what we are really doing is trying to create a better future, then that should lead many of us to question the very short-term oriented tactics which might not be as effective as we think.
These tactics could be simply based logic, inaccurate or misleading facts, or they could be arguments that simply appeal to people initially but may not have the lasting impact or long term "spreadability" that would be desirable.
It's in that last category of "not as impactful" that I include health and environmental arguments. These arguments are unfortunately commonly used, partly because of the common and unnuanced shortcut which causes us to over-prioritize the immediate, very short term impact vs. an approach that could be much more positive in bringing all of us to a more ideal world.