Let's say that historically everyone followed that advice. Then we would have no women's suffrage movement, because that movement upset many conservative people (including some conservative females) who strongly believed that females were insufficiently competent to vote. Then we would have no movement to protect the rights of workers to form unions and strike, because many conservative people (including some conservative wage workers) strongly believed that striking workers and unions were irresponsible and dangerous. And on and on. Every liberal movement against the status quo was supposedly destined to fail because by challenging the status quo such movements only anger biologically intransigent conservatives and are therefore counter-productive. Psychology and sociology scientifically demonstrate liberalism is futile and counter-productive, so don't even try, liberals should surrender and keep silent.
Absolutely, let's try to be reasonable and friendly and nice. But don't try to argue that psychology and sociology shows that liberals shouldn't voice their disagreements with conservatives. The fact is that public opinion does change. It does not change easily. It takes time and it takes effort. Change begins with a few people expressing unpopular truths. Anyone who thinks that popular and mistaken beliefs are going to change without argument, without debate, and without any of the conflict, risks, and inconveniences that this entails, isn't being real. Our civic responsibility is not to avoid upsetting people, our civic responsibility is not to express only agreement with majority opinion, our civic responsibility is not to be uninvolved, third party, objective, non-judgemental, observers. Our civic responsibility is to properly identify and favor the beliefs, policies, and laws that are better on the merits.
When we say we believe something exists we are making a claim about what is true or false. So merit in the context of belief isn't measured by how good it feels to have the belief, or by whether the belief expresses tendencies that were favored by evolution, or by tribal utility, or by anything other than whether the belief is more likely to be true than competing beliefs. Furthermore, the only method we have that we have any good reasons to conclude succeeds in justifying belief is best fit with the weight and direction of the overall available empirical evidence.
Being judgmental is an essential foundation for merit and ethics, and a refusal to be judgmental, an insistence that all epistimologies have equal merit, that beliefs are styles, that everything anyone believes is a product of evolution and therefore equally valid, that merit of beliefs is measured by its other effects, not by its veracity, is to sacrifice merit as the measure of belief justification. You are free to take that path if you want, but if you are someone who expects other people to adopt such an outlook, think again, that isn't a reasonable expectation. People are not going to stop arguing for what they believe is correct on the merits because other people are intolerant and react negatively. If you want people to stop arguing for their beliefs then you need to convince them to change their beliefs. Convincing me that people are intolerant of atheism isn't going to convince me to stop publicly arguing for atheism. On the contrary, the evidence that many people are intolerant of atheism refutes the argument that this is a trivial disagreement so we should focus on the other more important issues instead. I have said this before and I will keep saying it: If people were not so intolerant and fearful of atheism then fewer of the policy issues that secular humanists agree are important would be in dispute. The opposition to secularism is fed in part by intolerance and fear of atheism, so by confronting and challenging that fear and intolerance we are also promoting secularism more generally.