I agree, there needs to be a way to get them out. I'm just wondering if the rungs were not perhaps more beefy than what a typical ladder might be? Or perhaps they are recessed into the side? The plans don't seem to show that, they show rungs as you have them but I suspect they would be subject to damage from falling shot as it were if they were not beefy enough. And maybe they are beefy enough. Would 32 lb balls dropped from that height break off the rungs? I think it might but maybe it wouldn't.
On this note, were different caliber of balls stored in different shot lockers or were they divided to accept balls of a variety of sizes? The lower deck guns were normally larger than the middle and upper deck guns so it makes sense that in a line of battle ship, there would be several shot lockers, at least one for each size of shot. A two decker might have two different sized guns on her upper and lower gun decks but then there are carronades, chasers, upper deck guns (long nines perhaps) and each of those might be a different caliber again. How many shot lockers are there in a ship this size? There's a lot of weight of shot. A single broadside from a 74 given 28 guns on the lower gun deck of 24 lbs and 28 on the upper gun deck of 18 lbs would weigh 1176 lbs. A conservative estimate of 4 shot (likely more) next to each gun as ready use in the shot garlands amounts to 4 704 lbs of shot spread about the ship. And that's just the broadside guns. Doesn't count the carronades, chasers, AX/FX guns, chasers etc. Having shot for a number of engagements means literally tons of shot in the garlands plus much more in the lockers. It's a lot of weight to contend with. It's a lot of calibers to contend with. I recall studying battle ship evolution and the history of the dreadnoughts (I'm a naval officer so it's professional curiosity of course) and a big problem they had at first was with spotting fall of shot. Smaller guns can be worked faster so the spotter had to be careful about what shot he was spotting. There tended to be significant confusion about what guns were being spotted and that brought in the era of the "all big gun" ships so they didn't have smaller guns and larger guns shot falling in the same places and confusing the spotters.