Build a hypothetical post-apocalypse bicycle

Then I think we really are talking of cart level technology. Boneshaker - wooden wheels, iron tyres with pedals on the front wheel, add a sprung seat mount if you wish. Cheating? No need to reinvent the wheel when bikes were originally designed for exactly the scenario you are looking at.

^This. If you're going to reject anything that will ever wear out and can't be repaired by simply hitting it with a rock then we should just go back a couple of hundred years.

If we have more reasonable expectations that you might have a spanner and an allen key and an occasional chance to scavenge then obviously you'll want to standardise as much as possible e.g., 26" wheels rather than 700c because the tyres are easier to find.

The interesting bit, I reckon, is identifying components that will genuinely last a lifetime, like Royce bottom brackets, and maximising the lifespan of the bits you'll have to scavenge for. So on that basis it would be a fixed gear, chain-drive with full length chain case running on the best sealed bearings (BB and hubs) that you can find. Steel frame, as that could be welded back together (or go stainless). Your tubes will puncture and when you've run out of patches, stuff them with something squidgy, or go for solid tyres straight off. Front brake is a problem since all bike brakes wear down, if you went disc you'd save your wheel, but pads would be easier to replace or carry. Maybe a rod brake acting on a solid tyre would be the best option.