The above answers are fairly good.
A bicycle has to be lightweight but strong; the metallurgy to handle that is fairly recent. Cast Iron was heavy and brittle. Hand-working steel would be a major amount of labour, just to fashion one wheel. We don’t realize how much we put a modern bike through - the load on the axles, the forks; worse yet with uneven roads, the jarring and potential for breakage would be worse. Welding is a relatively new technology, so before that frames would have to be cast or bolted, meaning much heavier metalwork.
Anyone who’s ridden with a flat tire can immediately tell the difference an inflated rubber tire makes for a bike ride. Without tire technology, a bike would be much less comfortable, and the jarring and chance for breakage of the wheel and frame much worse.
A lot of the operational parts of the modern bike rely on ball bearings and grease, and then screw-on or press-fit pieces to hold those bearings in place. The easy free-wheeling of a modern bike wheel (which significantly reduces the work needed to ride) relies heavily on bearing technology. It wasn’t until the civil war that factories could reliably make pieces for, for example, a standard rifle so you could swap one piece for another. before that, most such tech was hand-fitted together so each product was a custom job. Precision metal-work is fairly recent.
Every wandered the older areas of an old town where they have decoratively left the old cobblestones exposed? Now imaging riding that on a clunky, heavy bike without inflated tires. Not fun at all. Out in the countryside, many roads became mud with the first rain and were almost impassable. Ever tried to cycle through deep mud? Roman roads were nicely drained, and paved with flagstones, but would still be relatively uneven and lumpy by modern standards.
Remember the silly-looking penny-farthing bikes, with those 6-foot front wheels? They were made that way because to get to a decent speed with direct-attached pedals, you needed a huge front wheel (or really fast feet). One of the biggest advances of the late 1800’s was bicycle chains, made from stamping and press-fitting fine metalwork. Making one of those chains by hand would be even more time-consuming for a master metalworker than making the wheels. Somewhat earlier versions had crank-pedal drive, where the pedals pulled rods that turned cranks on the back wheel like a steam engine - or the hobby horse (or Flintstone) drive, where you used your feet on the ground and coasted.
If a really good old-time craftsman were shown and allowed to fiddle with a modern bike, he probably could have built an almost acceptable substitute - minus the inflatable tires and chain drive. A fine glued laminate wood bike might actually work; you can see in the news every so often someone has a wood or cardboard bike as a novelty engineering design. But… would any such design stand up to anything more than smooth sailing along a very flat road?
We don’t realize the force we put on bikes. Consider how much stress you put on the pedal assembly by standing on the pedals; or standing up to pedal. Consider how deep the gouge in you calf would be if that pedal crank snapped in mid-run? What keeps the pedal crank turning the axle it’s on? one thin 1/4-inch bolt? That’s got to be some pretty reliable metalwork.