I was browsing the various sites on Stack Exchange and noticed some having a "Subscribe to newsletter" button.
Does such a thing exist here? Are there reasons why there isn't one, in the case there isn't?
Mea culpa, ELU does have a newsletter, and, as you suggested, it is automated.
You can subscribe to the EL&U — or any other SE site — newsletter here:
EL&U can be found under “Culture”, in the right column. You may have to click “show more Culture sites” at the bottom of that column. Or you can use the search field at the top left and type in “English”.
In case it’s of interest, I tracked this down by going to MSE, searching for “newsletters” (plural in the hope to find only posts about newsletters for all sites), clicked the first result, saw the tag newsletter, clicked on it, sorted by “votes”, and found this MSE post: Weekly site newsletters?.