Emotional tone is communication of values. If one does not allow people to communicate their values verbally, then they will communicate them in less productive ways. Like downvoting/upvoting people based on values. Even worse, it selects for people who argue without acknowledging their values. Even worse, personally, if one personally doesn't acknowledge their emotions and their value systems, then they risk hobbling their self-understanding. It contributes to the introspection illusion[1]. I think this happens a lot on the Internet in the 'rational discourse' communities (like HN, or LW, etc.): You can't choose not to be emotional. You can merely fail to express your emotions, which of course I believe impairs communication. After all, if I thought my own writing was objectionable, I wouldn't write it.
I think there is a similar trend in such communities (cf. Yudkowsky's Politics is the Mind Killer) to pretend to be apolitical, but insisting that one is apolitical merely obfuscates ones political values. Civil tone and apolitical politics are a popular sentiment amongst the type who seek to align their values with rationality.
You can be perfectly logical and also completely emotional. There is no antagonism between the two. Logic is emotional. One wouldn't bother writing out a logical argument without feeling it has some worth, right? Value is fundamentally emotional. Emotions are logical: They are responses and interpretations of stimuli that are predictable and behave according to rules that vary from person-to-person but have a common basis in all human beings. They have meaning, they can be analyzed and synthesized, and they can be communicated to others just like logical propositions and proofs.
Failing to communicate emotionally can lead to strong emotions on two sides of a discussion. One can think of it as boosting a signal to try to get it through. If one or both sides are unreceptive to the other's emotions, then this leads to a death spiral for the conversation. I think it is this phenomenon, the failure to communicate emotionally, which is being conflated with the presence of emotion. The solution isn't to maintain politeness and throw out emotions altogether, it is to learn how to spot breakdowns in communication and bring them into the conversation so they can be resolved.
Breakdowns in logic happen all the time. If one values rational discourse highly, then these are considered important to resolve, not ignore. God knows I've stood in front of a blackboard trying to get a mathematician to explain something to me, with the explanation failing to get through to me. I once spent 2 hours trying to construct the tensor of two vector spaces because I was a wee undergrad and constructing such widgets was new to me, but I had a patient grad student to guide me along. The breakdowns in logical communication were immense, and certainly there was communication of frustration and reassurance. Ironically in such a dry, logical matter, the emotional communication was excellent and the logical communication-- not so much.
[1] https://en.wikipedia.org/wiki/Introspection_illusion