If you remove the breech bolt (2) to check the firing pin(s), be sure to depress the portion of the rear firing pin (23) that protrudes out the bottom of the bolt (under spring pressure from part 25).
If you don't push that in, the firing pin isn't going to do anything useful.
...And that's one of the reasons why it was also suggested that you make sure the lever is fully closed.
In normal operation, the locking bolt (44) comes up underneath the breech bolt (2) as the lever closes. Just as the lever is fully closed and locked, the locking bolt engages the breech bolt to lock it in place, and simultaneously presses the rear firing pin (23) upward to be inline with the forward firing pin (22).
If any of the parts are not engaging properly, or the action is not fully closed, failure-to-fire is the usual result.
If your lever won't fully close, and no other problems are apparent (such as debris or excessive wear), check to see if the loading gate screw (43) is tight. A loose loading gate screw can bind the carrier (8), which binds the lever in-turn.
And, the easiest way to check for excessive wear that may have increased headspace over the years: (This is not a replacement for checking it properly - just a quick method for estimating.)
1. Make sure the rifle is unloaded and fully cleared.
2. Cycle the action.
3. Leaving the hammer cocked, press forward on the back of the breech bolt (2) and note the amount of movement. Anything more than about 0.005" movement usually indicates bad headspace, but you probably won't be able to see much unless it's 0.015" or more ... which is really bad headspace.
3.a. If you find it difficult to press against the spring pressure of the extractor:
3.a.I. Note (or mark) the position of the breech bolt and fully lower the hammer against the firing pin. Keep holding the trigger and press forward on the hammer. Watch the amount of movement in the breech bolt.
As sad as it is, it seems like most 336s built since about 1975 have left the factory with excessive headspace. The problem just isn't apparent, since the power of the hammer falling and striking the firing pin and breech bolt serves to push the breech bolt forward, momentarily closing the gap created by excessive headspace, and allowing the firing pin to initiate the primer.
I've seen 336s with over 0.035" of play in the breech bolt when locked. That's MASSIVELY excessive headspace. ...And the rifles operated without a hitch 95% of the time. They occasionally had light strikes, but a second hit to the primer usually worked.
Does your 336 have the cross bolt safety? If it is in the "safe" position the firing pin won't strike the primer.
Squeeze. Click.
Dinner runs away. Click. ???
"Oh, I..."
Been there. Done that.