How often do you consume some post-apocalyptic media and people slog it on empty highways or whatever for miles and miles on foot or revert back to fucking horses for some reason? Like I can suspend my disbelief that somehow there's just shittons of fuel laying around everywhere that magically doesn't spoil for decades but even then there's so many scenes where the characters are struggling to walk a distance or take forever and you could literally do it in a quarter of the time on a bike.
They're comparably easy to fix, they don't require keeping a whole ass horse alive, they're fairly abundant everywhere and the fuel is you. It just fucks with my disbelief. What, do these people walk through what remains of the suburbs, looting houses for supplies and just ignore every bike they see?
I have some bad news; horses are incredibly fragile, and if they break a foot or a leg, you're basically fucked and they're not going to be able to carry you again. Not to mention the need for a consistent food supply, and one that doesn't change rapidly to avoid colic. They can't puke, so anything wrong goes down that hatch and that could be the end of it.
I'd rather try to repair an inner tube with whatever random materials I can find around then try to do field veterinary work on an injured horse.
And you don't actually need bicycle tubes to ride bicycles. During WWII there was a rubber shortage and people got creative with makeshift tires made from rope and corks and stuff. It was a bumpy ride compared to inflatable tires but it worked.