You use them when level breaking plane. When you level break plane you use Zoll, plane blueprints and universal blueprints, you don't change them to plane blueprints. You buy plane blueprints from shop or use plane blueprints from challanges. Higher breaks use lot of universal blueprints so use them carefully.

When I try to level-break a cheap plane (5 plans) it'll tell me I have 0 plane plans and can't do the upgrade because of that, the status bar at the bottom shows ~11 universal plans. Do I have to do something to tell it to use universal plans?

No you dont use them that way, lvl 5 use only plane blueprints, so buy them from shop or you get couple for 5 games in that plane, next break will use 45 plane blueprints and 30 universal

Ah, so they're not wildcards for plane plans but a distinct resource.

yea, if you're going from LvL 5 -> LvL 10, you need to buy the initial resources. But from 10 -> 15 you use the gold ones that say universal, I got confuse too for me initially.

Cheap planes don't need universal plans until 15->20 either. Certainly doesn't help with figuring that out.