This isn't true. Many people bike on footpaths and surfaces that are completely unsuitable for motorised vehicles. Almost anywhere you can walk you could ride a bike and cross the terrain faster.

If walking turns into scrambling or climbing the bike becomes a bit of a hinderance, but they're not that heavy and the benefits across most terrain outweigh the inconvenience. In fact on a bike you could probably avoid it altogether, not losing a great deal of time just going around.

You're not likely to still be travelling long distances long enough after whatever event that the roads and paths are in such a state of disrepair. You'd be talking years for that to happen and you probably would have had to sort out some sort of place to live and grow food long before you had to worry about it.