Machine bows should definitely be possible to build for a medieval engineer or also a Roman engineer. Not hand held, but stationary and more artillery-like. You could position it on top of a wall and attach a heavy weight. Letting that weight down powers the mechanism, just like pendulum clocks power the clock. The arrow - or even just a stone - falls down from the magazine into the "chamber" it is then propelled forward, the mechanism pulls the string back and the cycle repeats until the weight has reached the ground and a new one needs to be attached. That would have increased the efficiency quite a bit, I would say. Was there no use case for such a weapon in medieval warfare?
Hey there,
Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.
If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!