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 emulate modern spoke systems with sturdy leather strands. The chain is difficult point - leather won't cut it as you need immense tensile strength and rigidity - but if you didn't want to meticulously hand-craft a modern chain (possible but tedious), a direct-gear solution might be practically possible.
No, what was missing was the very concept that it would be a worthwhile thing to attempt to build. It would have taken an incredible visionary to follow the above steps, especially in a world where the poor quality of roads made such a device of questionable value. And anyone rich enough to afford such a thing was doubtless rich enough to afford a horse carriage - who in that set would aspire to pedal through the mud?
tl;dr expensive, flimsy bicycles are completely useless in a world with poor roads and deep wealth inequality.