Your other option is to take the drives out of the enclosures and stick them either in a dedicated NAS device, or into a home server or something similar. You could also plug them into a server's USB ports, but I think keeping a bunch of USB drives around probably just generates unnecessary clutter if what you're looking to do is use them as permanent storage.
As far as how plug and play the USB-equipped routers are: generally, no configuration is necessary. That's also true of any computer you'd use as a server too, though. I would be 100% confident that any router would recognize a plugged-in USB drive.
The main limitation you should be aware of is drive spin-up times. USB drives typically will be shut down when not in use (most modern drives spin down on their own - if yours doesn't, the enclosure or the router both may spin the drive down as well).
Another limitation: USB issues. USB drives are going to be somewhat slower than internal hard drives. And, if you have several running on the same bus, you risk saturating the bus and slowing down all of the drives as they become starved for bandwidth (of course, this depends on your exact usage and how much data you're cramming through the pipe).