If you're uncomfortable with the ladder's installation after following the manufacturer's instructions, then I suggest a low risk proof test: hang it out the window, take the stairs down to the bottom, and put a bunch of weight on it. What's the proof test load? That's an interesting question in that I assume that this ladder is for escaping fires instead of drug raids. In the fire scenario, I can easily imagine your ladder looking very inviting to occupants of the two apartments between you and the ground level. An upper bound on the design weight, then, is me, any family, any dinner guests, and all of my downstairs neighbors and their dinner guests. With the height constraint of 4 stories, 5 average weight people sounds like the most who could descend at the same time. Be careful not to break the bottom rung of the ladder with proof test weight. Weight in the center stresses the rung itself, whereas weight on the ends just stresses the connection to what I assume is a rope support.
Alternatively, wood connection design is unfortunately the most difficult from among common construction materials. The National Design Specification (NDS) is the US authority on design values for fastening into sawn lumber like wall framing (although other materials than sawn lumber do get used in wall framing). Chapter 12 covers dowel-type fastener design (nails, wood screws, bolts, and lag screws). My mental image of your use case is just a ladder hanging from a bar, so I suspect that withdrawal strength isn't relevant to your project, where instead you want to compute lateral strength. If there's a weak layer of stuff between your brackets and the wall studs, however, then the NDS's lateral load design values will be no good (withdrawal loads would still be okay, though).
The institutional structure behind the NDS (the American Wood Council) has a technical report providing lateral load design values for my "weak layer of stuff" cases like window trim, but it's near incomprehensible for non-engineers. I don't know of any online calculators that cover the scope of that technical report. The technical report provides the fundamentals to answer your linked StackExchange question correctly (or compute a strength given an intermediate layer of poorly fastened trim), whereas the NDS isn't up to the task; that 300# is total BS.
You're right about wood density being a common variable in fastener strength computations. Technically the NDS uses specific gravity (with "G" as its variable) instead of density, but they're effectively the same thing (you multiply specific gravity by the density of water, (62.4#/ft³)(1ft/12in)³, to get density).
Among the common US softwood species used in construction (Douglas Fir, Hem-Fir, Southern Pine, and Spruce-Pine-Fir), Spruce-Pine-Fir (South) (or SPFs, where the little s is for "South," not plural) has the lowest specific gravity (0.36 from page 87 of the NDS's chapter 12), making its fastener strength values conservative when working with sawn lumber.
For your single shear connection, there are 7 failure modes visualized below from the Appendix chapter of the NDS (page 184). Standard practice is to compute all 7 and take the smallest as the strength. If you're anchoring through a piece of wood trim and want to use the NDS instead of that technical report, then you would need to attach the trim sufficiently so it acts as a structural member. It can't shift before the ladder's fasteners reach the smallest failure load among the 7 modes. Given such structural trim, you can pretend that the framing thickness is the actual framing thickness plus the trim thickness. I really don't like this strategy, because for all I know your trim is some engineered wood product without even the strength of SPFs. And I've seen trim weathered into dirt, although maybe that's an unreasonable expectation for a structure with 4+ levels.
Assuming that the trim is at least as strong as SPFs and attached sufficiently, you can use the NDS to compute your fastener design strength. Section 12.3 of the NDS is where to find the lateral strength prescription (starting on page 83). The 7 failure mode computations are nasty, but don't worry about that. The American Wood Council maintains a connection calculator that will compute them. Take note of Fem and Fes in the expressions, though, and observe Table 12.3.3 on page 86 to see how the specific gravity plays a role in the failure mode values. For the connection calculator, my instinct is that your ladder has thin brackets. I would use steel for the calculator's side member, choosing its thickness based on the bracket's thickness. Use load duration factor CD = 1.25. Depending on your climate and rain exposure, you might use wet service factor CM = 0.7 to account for high moisture. The other 2 factors in the connection calculator should be 1.0. When in doubt, use Spruce-Pine-Fir (South) as your lumber species.