I really appreciate the discussion and idea for fastening and gluing the rungs for this ladder. Thank you all. I will go with the construction wood screws and construction adhesive. I know I've seen screws like that at the local hardware stores but I have not looked at what sorts of construction adhesives they carry. I know the brands of screws Stick posted are NOT available locally, but one brand is at one store 50 miles away. Because of the rigid mount, I may not go with gussets.
So, moving on ...
I am thinking ahead on how to hold the rungs square as they are fastened/glued to the first stringer. I am thinking all the rungs must be secured to one stringer first and then the other stringer added in a second stage. It is possible the rungs would mount fairly square without any jigging (is that a word? - sounds like a fishing action) because the pockets have been routed square bottomed and the ends of the rungs have been cut very square. But, I do not want to rely on that, solely. I would like to come up with a jig, large squares that can be clamped to the stringer and rung while it is being glued and screwed in place. I have a couple of the large construction squares and they clamp up okay with the stringer flat and the rungs sticking upward.
However, that is not a position in which I want to be gluing and screwing, from below. I am not confident that those squares will stay in place if the clamped up rungs/squares were laid over flat, so I think some blocks or something need to be attached to the squares for more secure clamping. I am okay with drilling holes in at least one of the squares for attaching blocks. Alternatively, it could work to clamp blocks to the stingers/rungs and then clamp the squares to those blocks, eliminating the need to drill and bolt blocks to the squares. But... that is a couple extra steps that may not work well with glue setting up.
Thinking further, glue and screw the two first rungs while laid flat, without clamping square. Then tilt it up so the rungs are upright and then clamp them square (as pictured) and let the glue set up. That could be done quickly and well within the glue working time (if not, I need to find a different glue as the final glue up of the second stringer is going to take much longer).
Follow that same procedure for adding the other rungs and use a lath to tack the others square, using the clamped square to align each rung, prior to tacking.
I am also starting to think about the entire gluing/screwing operation/sequence of once the rungs are all in place on the first stringer, how do I go about adding the second stringer and gluing/screwing all nine rungs? Will the working time of the adhesive allow for applying glue to all the pockets/joints, press the stringer on and then drive the screws? I am thinking that at this point the assembled first stringer and rungs can be laid flat on a bench or horses so the screwing of the second stringer could be done horizontally. I anticipate that holes for all the screws will have been pre-drilled in the stingers and the screws in place ready for driving into the rungs.
... OPEN THE FOOD GATES !!!