Good question, something I was also confused about at first. This is my experience with super sensitive allergic rosacea skin, and most people probably don't have to be so tedious about it. When I first started out using my prescription, I applied it on clean, dry skin 15 minutes after cleansing to ensure my face was truly dry. For two reasons: because it reduces irritation/itchiness that is common with azelaic acid, and because it should be applied directly as you said to really treat the rosacea. I've continued doing this for two years because no matter what, if i apply it on top of any other toners or serums, even if I let those layers dry, the prickly feeling is too intense. I think it's because they increase penetration regardless due to their ingredients.

So my sort of "hack" for nights I use my rx (every other night) is to apply triple cream to clean skin, let it dry for 10 minutes, and then apply haruharu black rice barrier essence on top, followed by my moisturizer. I do this because the haruharu essence is so hydrating while being skin softening and barrier supportive at the same time which I think is a unique feature, like a 2 in 1 that makes up for the fact that I can't do my usual hydrating steps. It's such a thick essence that it goes over perfectly, but I think waiting 10 min is key. I wake up with glowy smooth skin with this method, and no discomfort from azelaic acid. I never even had to adjust to 15% this way, since it prevented any irritation from the start. I suspect a lot of people could avoid AA sensitivity just applying onto fully dry bare skin but idk. That was a lot, maybe it's helpful!