You know how sometimes it's easy to get carried away searching for the perfect product to end your woes? I initially did that with this cabinet. I tried a cute rail to mount on the door and store pot lids. It felt like the lids were going to tumble every time I opened the cabinet with the least bit of gusto (and I often do open this particular cabinet with a healthy dose of gusto). I tried a wooden rack for storing pot lids, too, but it took up too much surface area.
Turns out, I was really most comfortable with the simplest solution here.