How are higher speeds? 1/60, for example?
You are on the right track, most likely. Bending that pin is the answer. And it can be very very touchy. I will cock the shutter, move the speed cam toward 1/30 to get the pin to slide down the ramp a bit, and grab it with smooth tipped needle nose pliers. Then give a slight bend one way or another. Keep the pliers on the upper straight part of the pin, not grabbing the wider base flange. If I try to bend without cocking the shutter, the pin is too close to the outer edge for me to get much movement.
'Much' being very relative here! Small small bends. It is possible to break that pin off.
When the shutter is not cocked and you move to 1/15 second, then cock the shutter, you should see that pin move a very small amount. Just a touch. If it is hitting the speed cam before cocking, it will be too fast. If too far away, too slow. And we are talking in 1/100s of a millimeter. I find watching the little movement toward the came that shows me how far off the cam the pin is is a good check on things.
There are two springs in the escapement that are prone to come off their settings. At each end for the two dark gray parts that curve down. But it soulds as if yours are fine.
Patience and more patience.