Addon that's been around forever (and regularly updated) that keeps track of instance run times (among other things):
InstanceTime.
Can announce to various channels, keeps track (optionally) of loot dropped + reputation gained and so on. And keeps the runs in a database you can easily access later.
It doesn't keep track of "average run times", but a calculator/notepad+pen can do that with a collection of run times. Another thing that can be done with a calculator/notepad+pen and the run times recorded by the mod is "
Calculate the average time spent per badge from each instance".