I think the problem was roads. If you combine bad (or nonexistent) roads with airless hard tires, the result is much much less useful than any modern bike on any modern road.
The first bikes were indeed wooden with wooden wheels. No ball bearings, no hollow steel tubes etc. But streets.
I could now say something about how engineers are so specialized in their perspective they cannot judge things without bringing current conventions into it but hey, every profession comes with it’s weakness.
Highly skilled artisans could make one-off pieces fit together exactly. Doing so at scale was prohibitively expensive, because machine tools hadn't been invented that let you precisely machine things identically. So, if you had a broken bicycle, you'd need a skilled artisan just to repair it- off-the-shelf parts were impossible. Every single bike would have been a unique object. This is how guns, ships, clocks, and basically everything was made before reliable methods of achieving precision were invented at the dawn of the industrial revolution.
The antikythera mechanism had inherent looseness in its hand-wrought gears which greatly limited how accurate it could have been. https://en.wikipedia.org/wiki/Antikythera_mechanism#Accuracy
(source: this book, which I gather is pretty accurate, if pop-history https://smile.amazon.com/dp/B072BFJB3Z )
Ancient technology was neither accurate nor precise. Making two specific things fit (e.g. THIS dowel pin in THAT hole in a brass plate) does not require precision in the actual sense of the word, you can achieve the goal by consistently small iterations (ex: grinding the dowel down until wanted fit is achieved).
Precision (all my threads are the same) and accuracy (and they fit with the thread from every other machine shop) are what made the industrial revolution. Ancient technology had neither of those.
Dually, Romans were known for their extensive roads system. My family home sites directly below the Ancient Roman frontier in Germany, where you can climb a mountain and still find a 10 foot wide stone road they built 2000 years ago mostly intact
They often use customized bikes for that to keep from shaking their fillings out.
There was basically no rich middle class which had the time and money to fool around with individual transportation, and the other classes had either no real reason to want this or it was pure luxury.
I also would think such people existed elsewhere in the world centuries ago.
Chinese wheelbarrows are the real endpoint of pre-industrial transportation.
Pretty much all modern rideshare bicycles do not have pneumatic tires, FWIW. Ride quality suffers, but it's not an awful tradeoff for short trips with whatever squishy compound they use now.
Carriages used to have suspensions (made from some elastic whale bone I believe).
If someone would attempt to build a bicycle in a time of carriages with carriage wheels, I would then expect this person to also copy (at least in analogy) the suspension systems for carriages of that time, if not on the first try at least on the second...
It's just some of the cheap dockless purveyors that use non-pneumatic tires, and these companies have been dropping like flies. The docked bikeshares by contrast are still (slowly) expanding with their superior, but admittedly more expensive, bicycles.
Anyone with the means to pay would much rather use an animal-pulled cart instead of literally making an ass of himself.
Or just ride a horse like a gentleman. (Until the 'mad dogs and Englishmen' craze hit.)
As a counterpoint see the marine chronometer for measuring longitude. Clocks had been around for centuries so easy right? 
Making something strong/reliable is easy, making something precise is 'easy', making something strong/reliable and precise, oh and preferably lightweight is hard.
You do have a good point about roads though. In fact in the UK the first push to tarmac the roads came from cyclists, not from motorists. And the first tyres were for a certain Mr Dunlops sons bicycle.