The Hunter One-Trigger is a night-mare to work on. There are small springs, a tiny ball bearing and quite few moving parts. The worst thing is dirt and if someone had it apart and oiled it too much, now the oil solidifies and causes problems.
Still once you have one in front of you and look at it, it is a marvel of engineering.
In Brophy's book "Plans and Specifications of the L.C. Smith" it is broken down and most of the parts have drawings with specifications.
Hunter One-Trigger

Here is a photo of why the safety is manual instead of automatic.

From the photo you can see that how high the spur lever has to travel for the second shot. The firing plate has to travel that high to reach the other sear.
So they had to mill this area out and along with it went part of the groove for holding the safety push rod in.