Ability to show basic statistics as a function of time

Yes this'd be nice. Here's the new-comments-per-day, and new-pages-per-day:

(Let me know if you want more SQL for sth else.   & Sorry for the a bit late reply.)

select
    date_trunc('day', p.created_at) as date_trunk_day,
    count(case when p.post_nr > 1 then 1 end) new_comments,
    count(case when p.post_nr = 1 then 1 end) new_pages,
    s.id site_id,
    h.host
from posts3 p  -- will be renamed to posts_t some day
    inner join sites3 s on p.site_id = s.id
    inner join hosts3 h on h.site_id = s.id and h.canonical = 'C'
where
    h.host not like 'test-%'
    and p.post_nr >= 1  -- ignore title (nr 0) and special posts, e.g. bookmarks (nr < 0)
group by
    date_trunk_day, s.id, h.host
order by
    date_trunk_day desc, s.id
limit 365;

extract this type of data and visualize it ourselves.

How would you go about doing that? (For example, how do you "create" an API to get the data out of the database?)

I did experiment quickly on localhost some time ago with connecting Metabase (I think it was), https://www.metabase.com, directly to a Talkyard database. Was pretty easy to get it working and showing different graphs. However, in your case you'd need some authentication in front of Metabase.

Edit: There's also Apache Superset, https://superset.apache.org, and Redash, https://redash.io/. (And more ...)

We could then always feed this to various graphics engines

It would be interesting to hear about what tools you're using :- )