Sketchy Old Bicycles

I visited my grandparents today and Granddaddy rode his bike again, but this time Hope and I were with him. He began shouting, "Oh, cuss! The brakes don't work! [Ernie of Bert and Ernie from Sesame Street laugh]." Hope and I ran after him and caught him at the edge of the pasture, and he had that grin on his face that totally said, "Children, that was a Shady MacDougal for the brakes DO work! Hee hee hee!"
And I finally figured out what model his bicycle is. It's a 1972 Schwinn Collegiate without the chainguard. Either the bike was sold like this or he decided he didn't need it.

We also took my grandmother's bike back to my house. Unlike my grandfather's bike which had oxidation and some light pitting, I think my grandmother's bike needs a complete overhaul. Tomorrow I'm stripping it down to it's frame to make sure it's still safe to ride, and I need to degrease EVERYTHING. The most curious part of the bike is that it has a rear wheel hub that has a small chain coming out of it. I wasn't sure what it was; I guessed that it was just some sort of coaster brake, but that seemed strange to me because it has rear and front caliper brakes, and a shifting lever. But, I have absolutely no idea how the shifting system could actually work considering there's no cassette, no derailleurs, no jack squat. It would be kind of funny if by shifting into higher gears with the Archer shifting system that you'd engage some sort of flywheel which would make going up hills easier, but that's just the inner dork in me poking its argyle sweater clad body around the corner.

UPDATE: So it's not exactly a flywheel, but the rear hub is actually an internal gear hub as shown in the diagram that uses epicyclic gearing. I guess shifting by pulling on the shifting lever would change the gear out of a low input gear with large secondary output gears, to a high input gear with small secondary output gears. I'm not absolutely sure how the gears would engage and disengage, but I've got a feeling that there might be some sort of ratchet and pawl type system that would allow all the gears to push together and lock together when you have the highest gear selected, and all the gears separated when you have the lowest gear chosen. I'm not sure; I'd have to take it apart to see how it works, but the only advantages it seems to have is that you'd be able to shift while standing still (which you can't do with a derailleur system) and because it's sealed, you wouldn't have to worry about dirt and grime causing accelerated tooth wear. The disadvantages are that a bicycle requiring more gears and cogs (such as a road bike) would have to have a lot of input and secondary output gears for the rear hub which would end up being pretty heavy, and it would be hard to actually take it apart and put together again if you needed to replace a gear.
And the thing that blows my mind? This type of gear system was developed in 1898 according to Wikipedia. That's 7 years before a derailleur system was developed. I'm not saying it's more advanced than its rival, the derailleur, but it's certainly more complex, and more intricate, and involves more moving parts. What kind of guy would look at a bicycle and think, "Hmm, if I had gears apart from that measly cog at the rear wheel, then I could go anywhere!" and then devises a complex device to engage different types of gears enclosed in a tube? Seriously, if I'd been there, I would've gone, "Dude! Just put a bunch of chain rings on the bottom bracket and cogs on the rear wheel and devise some sort of simple lever that will move the chain from gear to gear!"
Once again, this is a perfect example of simple being better.