Expected moderation style is pretty community-dependent. In my experience it's been the other way around predominantly. Smaller communities can very much entangle moderation and the overall emotional balance among the members, especially when there's more delicate situations involved. Even in larger communities, depending on the interaction mode, moderators can be implicitly called upon as mediators, which involves a lot of trying to avoid people feeling like your response was insulting or unfair even if it was, and conversely making sure they feel like their own needs won't go unacknowledged.

In larger, topic-oriented, and/or highly generalist communities, ones that are looser and more impersonal in general, it's been more as you describe.

And if you are a good surgeon for example, you don't let emotions interfere with your decisions and actions during surgery either, but a mother of three dying on the operating table, with nothing even the best surgeon in the world could have done better, still sucks, while seeing a family break out in tears of joy when seeing a patient again after a surgery that went well, that still feels great.

There's a difference between handling something professionally, or not endulging in it, and not feeling anything at all.

Exactly. Which is WHY it is emotional labour....

From Wikipedia....

"Emotional labour is the process of managing feelings and expressions to fulfill the emotional requirements of a job.

More specifically, workers are expected to regulate their emotions during interactions with customers, co-workers and superiors."

The mods are expected to regulate their emotions.

Which makes it emotional labour by default.

What are your motivations for arguing this?

By this definition my response to you is emotional labor and you owe me for that.

It's not really, no, unless it's your job (or "job") to write this reply.