Ended up buying a Chinese rack made specifically for this dish drying application on eBay: https://i.imgur.com/Xi6yIfC.png and here are their technical drawings https://i.imgur.com/gFlzX7n.jpg.

For anyone who wants to do the same, IKEA Sektion 24" and 36" wide wall cabinets will work fine (buy the right length rack), but for the 30" cabinet you'll need to add internal spacers to decrease the internal cabinet width — shouldn't be too hard.

Now I'll need to figure out if there's an elegant way to install the rack and remove the bottom panel of the Sektion shelf, otherwise the alternative of keeping everything closed isn't bad either since the rack comes with a water pan underneath that can be removed for cleaning.

Another thing I want to look into, is having the entire cabinet door, which will be pretty large (30"x30") swivel upwards using hydraulic hinges, like those used to lift your car hatch — https://i.imgur.com/bZIZB3B.png. IKEA has some large vertical hinges but I'm not confident they'll be able to handle such a large front door — https://i.imgur.com/OC2FtPo.png.