It might be the cocking rod(s) that are hanging up inside the frame. MAybe they are being held back a little by the trip in the ejector system.
They run from the front to the back of the frame (thinking it's the same as the NitroSpecial).
Checking the firingpin strike w/a wooden block I take it you have the bbls off.
The forend is also removed of course.
The hammer(s) can drop and when they do they are supposed to push the cocking rods forward and push the cocking arm out.
With the bbls and the forend off, perhaps any resistance is low and the hammers can drop and strike the firing pins and leave a nice imprint in the block.
Putting the gun back together and trying it with live ammo,,now you have the cocking rod being pushed forward as before and kicking that cocking arm out.
But if there is anything in the forend shoe or the ejector mechanism that is offering resistance to that motion,,that will buffer the hammer fall and make for light strikes on the primers.
The problem may be in the forearm/ejector mechanism. It not allowing the cocking rods and arms to easily push forward under the fall of the hammer(s). With the forend off, any resistance from that is gone and a nice hard strike is seen on the wooden block.
Fore-end on and gun assembled, light strikes on the primers.
Try firing w/live ammo with the forend removed. See if it fires or if you still get light strikes on the primers.
Place the forend back on before openiing the gun so the hammers recock and the firing pins don't drag on the primers when opening it.
I'd take the cocking arms out of the front of the frame (cross pin) and mark them R & L for reinstall.
Then Pull each of the cocking rods straight out.
IIRC they are not held in with the double hex nut at the front like the NID.
They should come out easily. If not bounce the front of the frame on a lead block sharply to encourage them to exit. Sometimes it takes a bit of that and picking and grabbing to pull them out.
They could be slightly bent, the ends may be peened,, or both. Not uncommon.
Check those cocking arms for peened over edges where they engage the rods. That may be jamming them in their slots in the frame. Everything needs to move freely.
There's no precision fitting in these, just parts that slide and pivot.
The hole the rods occupy each may be packed with crud not allowing free movement, that needs to be clean as well.
About all I can think of for now.
No need to take the hammers and mainsprings out ,,yet,, as the strikes seem to be OK on the wooden block.
They are a tough one to reinstall w/o a jig and a drill press (again assuming it's the same as a Nitro Special.