1. It's remarkable how quickly bicycles can break down, especially when they're not made to be ridden (US department-store bikes, all Argentine bikes) and when you repair them with improvised tools and improvised or substandard components.
2. There are a fair number of specialized tools that you need for bike repair that are pretty hard to do without. You can maybe cut a cone wrench from a flattened spoon, and you can "press" out chavetas (cotters) with a hammer and chunk of steel instead of a cotter press, and you can true your rims on the bike if you have to, but how are you going to unscrew your freewheel without the right shape of freewheel extractor? Or break your chain without a chain breaker?
3. A bit further afield, you have to patch or replace your inner tubes when they spring leaks. But it's increasingly difficult to get working vulcanizing fluid for patching after the economy collapses; that shit doesn't have an indefinite shelf life. Ultimately you need to put together some kind of chemistry lab, using chemicals of unknown quality, in order to renew your supply of bike patch kits.
4. More generally, bicycles, like other industrial machinery, are allergic to entropy. The crucial resource needed to repack your wheel bearing balls isn't the steel to make the balls or the dead cow to grease them with; it's the knowledge about which balls are the right size, and which ones are already subtly cracked. Buy them in bulk and they cost a fraction of a cent. Install a ball of the wrong size and you could wreck your cones or your hub or even your body.
So I think bicycles are an important part of post-apocalyptic transportation, but keeping bicycles running without an economy populated with specialized equipment, materials brought from far away, and people with specialized knowledge could be quite challenging indeed.