The Reason Why Bard Is Not Picked As Much...
...is because he amplifies all of the game mechanics you are bad at. If you cannot evaluate the situation correctly, you will probably leave to get chimes and your ADC will die. If you cannot position correctly, your Q is a slow skill shot with ok damage and a bad slow. If you cannot figure out where your ADC is safe, you will place shrines in all the wrong places. If you don't know the current cool downs of enemy and allied mobility spells, your E could be saved or used at the wrong times. If you don't know when, where, or who to engage on, your R will kill your team, as in actually kill them.
You need to be good at so many things as Bard and if you aren't good at all of them, you will be bad at Bard. And when Bard is bad, his team will be bad and you will lose. That is why Bard is not played. An interesting thought I realized recently. Also a call out to other Support players that, if you are good at positioning, shot calling, and memory (remembering cool downs) then you should try Bard because he is really good if you are good at all of these things.