How helpful / beneficial this would be. What use does it serve? Does it make Hubski a better place and push us to be more thoughtful?

The benefit would simply be to the curious. It serves to give some indication on the flow of users to the site. If, for example, conversations change or new names start popping up, we could look at the counter to see if there was a spike. It would also give us an idea of when internet drama is happening.

I'm not sure if it would make this a better place or make conversations more thoughtful, but it would be a new layer of transparency.

Are there any unintended consequences? Do other sites have this feature? Why? What does it add? What does it subtract?

If it's a slow day and some people come to check out the site for the first time, they may be alarmed by low numbers or think the website is a "dead space". This could be good if it scares away those looking for popular opinion, bad if it scares away others.

Reddit has a feature that shows how many people are in a subreddit at a given time. I don't know if I really trust their numbers though (they've always had a thing about appearing larger than they are). I don't know of many smaller social aggregators that have a current viewer count, I think mostly because it's always in their interest to seem bigger than they are.

Would it break the site / unduly burden the site?

That one I couldn't tell you, I'm no developer. It seems like it's not exceptionally complicated, but I have no idea.