However, bit.ly provides nice click stats for each shortened link (example). Is there anything similar available for t.co shortened links? (I've tried the bit.ly way of appending
+ to the url to get to the info/stats page, but that, at least, does not work.)
Doesn't look like Twitter provides any additional services other than URL shortening, but this can change fast. In the meantime, I use the Google shortener goo.gl - great analytics, same as their usual web analytics. It uses your Google ID btw, although you can use it signed out (and give up the analytics)
Go to https://ads.twitter.com/ and login using your Twitter account. Go to Analytics → Timeline Activity.