On the one hand, it is a decent way to show readers how popular a post is and maybe feel more encouraged to share. On the other, now that I don't receive that count portion of the service, keeping the button installed means Twitter gets to still track my users.
People do use the buttons though to share. Before I added a LinkedIn button I received a number of emails from people asking for one so they could more easily share links.
I may just remove it though because when Facebook and LinkedIn have a good amount of shares or likes, it looks strange to have a button with no shares.
I've observed a number of times that the most popular web sites and the most popular news sites (as ranked by web traffic) are the most controversial (eg. sexist) and least informative. Hence, trusting the average person is like trusting populist politics.. The Kardadhians rule. Aka race to the bottom, aka McDonalds journalism.
And I agree, it is interesting to see it - and obviously Twitter would have that data, so they could make it available to researchers as well. On a daily basis I wonder how it impacts everything.