Biggest changes here are to try and mimic the reference footage more closely in terms of angle, shuttlecock spacing etc. I altered the angle that it initially lands at so that it sticks out slightly. It should probably bounce off in that direction as well, though - at the moment it just sort of goes in a straight line which looks a bit weird.
I also once again shortened the distance the shuttlecock travels after its first bounce and increased the height. I think it looks a lot better this way, but the second "bounce" is bothering me a lot. If it lands on the feathers it shouldn't really bounce very high or very far, if at all - it really needs to land on the feathers, then come straight down onto the cork. However this is very tricky to animate owing to the construction of the rig - the shuttlecock will always rotate from the cork which means that if I try to rotate the feathers upwards the cork will stay at the same height. I could probably simply adjust the height manually - thinking about it, it sounds very logical and very simple. I don't know why I've been having difficulty with it. Maybe I'll give it another shot.
I also removed the series of smaller bounces at the end. The shuttlecock comes to a very sudden stop after the feathers drop back to the ground now, which I don't really like - I think it should probably bounce very slightly once more afterwards. It's a bit too slow as well - once I start adjusting the curves I can remove some of that cushioning/easing.
I also managed to break yet another rig - look closely, once again you see that the translation control has detached itself from the ball of the shuttlecock. I have absolutely no idea why this keeps happening. This time, I almost certainly did not place any keyframes on the wrong object - it seemed to occur after I started Ctrl + Z/undoing a set of keyframes that I didn't like. The control object just seemed to pop away from the rig. Very, very weird. And annoying.
Please try to see past the ugly capture...! I've started adjusting the curves at this point, giving the shuttlecock a faster bounce and a slight delay at the top to allow it time to rotate downwards.
Looking at it closely, I think it probably hits the peak a little too suddenly - almost like an invisible ceiling. I think it needs to delay perhaps just a little longer to allow it to rotate a bit more before it drops.
I tried speeding up the fall onto the feathers at the end but it still looks too snappy/choppy. I really don't know what to do about it - my main idea for a fix involves getting the ball to drop straight from the feathers onto the ball and back again, in one place, without traveling forward and backwards or up and down or jerking all over the place.