Then I just tried to stick one of the old floppies in a Mac. And I amazingly saw it mount just fine on the Mac's desktop, with a ProDOS icon and all its files.
So it seems that my floppies aren't dead, but my drives could be (both of them, I tried to switch them to no avail). The eject mechanism do work, as do the red activity LED on their front.

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Not all ProDOS disks are bootable. Even if formatted correctly, unless they have files named PRODOS and BASIC.SYSTEM on them, they won't get past the "not a startup disk" screen. Mount a few in the mac and see if any have those files. If so, try to boot them in the IIgs. If _that_ doesn't work - then the IIgs drives must be dirty or otherwise failing.