Earlier quoted context omitted.
Good steel would only be strictly needed for axles. Bearings could have been polished and oiled wood. The frame, wood. The hinge of the front wheel, brass or another small bit of steel. Transmission could be a leather belt. Wheels, wood again. Would such a device somehow emulate a modern bicycle? Very likely. Would such a device be practical, reliable, fast, to say nothing of comfort? I greatly doubt it.
Bamboo-frame bicycles exist today as a fashionable item, and as far as I can tell without having ridden one, they work for practical purposes just as well as metal-frame ones. I suspect you could make a perfectly acceptable modern bicycle with 18th century technology, albeit not as durable. Metal axles aren't much of a problem. Light and strong wheels are trickier, but you could get away with solid wooden ones, or em…
Leather is good enough for motorcycle drive belts. Nowadays belt drive bikes use cogged composite belts, but once upon a time they were leather.