Good post. I think there are two ways of looking at this: 1) what's good for core UI change, and 2) what should be relegated to extensions, if done at all.
Core change
One thing I'd like to see is more distinctive handling of topic tags from category (organizational) tags in the All Discussions view, instead of just one long tags list with no distinction but color alone (which isn't distinction enough). The colored category tags are fine, but maybe the topic tags could be an old-fashioned tag cloud, or something. That would not only make them distinctive from the categories tags list, but it would save space better too and reduce the length of the side bar.
I'm not crazy about tag clouds, mind you, but something is needed there to improve on that distinction.
Widgets
These should definitely be extensions if done at all. I think Flarum has done a great job so far with the UI. Additional content features would just distract from the purpose of the system, which is to discuss latest events (more or less).
To elaborate on this a bit, consider the ideas mentioned: "most viewed", "most replied", and "most liked". I find stuff like this needlessly distracts people (which might be a marketers intention), leading them outside of their immediate objectives (not really catering to why they are there).
Take most replied, for example... There are many situations where a discussion might have a lot of posts, but doesn't need to be resurfaced again because the topic is now irrelevant. Whether it's from 2007 or last month, doesn't matter; if the topic is irrelevant to the time at hand, you may not want to be bringing attention to it.
Most viewed is another misleading metric and counter to the 'sink when dead' nature of discussion boards. We have no clue why anyone might click on a post. Hopefully it's because posts have real value, but usually that's not the case, and especially as time goes by. People are kind of like curious sheep... you put a "most viewed" label on something and they'll click and see what it is just because there's a label on it. All those false clicks augment the "most viewed" algorithm; the posts remain being most viewed but not for the reasons you'd want them to be. Marketers love curious sheep because they're so easy to fool. How many times have you clicked on something called "most viewed" and felt ashamed for clicking it afterward? ?
Most liked? Well, that's a very subjective thing, isn't it? And, again, giving special attention to such posts may not be catering to the immediate tasks/needs of boards users. However, if this was an extension, I might use this one to help show to new arrivals what kinds of things people are valuing. On the other hand, it could introduce a needless "gaming the system" behavior, where people start making silly posts just to get likes and listed in the widget. Depends on the boards' audience and posting rules, I guess.
