I think that it would be a game changer in naval warfare, maybe even up to the early musket era. During ship boardings, fire rate is very important, and once you start closing in, so having someone draw a low-powered version of these at 50m and pump several shots in a row might be more practical than loading a black powder weapon. And while it's more complex and prone to failure than a regular bow or crossbow, when your high at sea you have more down time to fix things.
This kind of weapon might also beat a crossbow when it comes to self-defence when travelling. While it's bulkier than a crossbow, you can always keep it loaded without causing any damage to its mechanism, whereas if you keep a crossbow ready to fire at all times, you'll be putting a lot of stress on the limb.
While it might not be effective for volley firing, it's definitely an advantage for skirmishers looking to ambush the enemy. A small group of IL archers emerging from a valley can wreak havoc on a platoon of soldiers marching downrange. Killing as many enemies as quickly as possible is fundamental in such a situation.
Another situation when it might come in handy - defending against an assault on a defensive structure. Pop out from the crenelations, fire your magazine away, pull off, reload and repeat.
It also seems like it might be a good hunting instrument. Maybe a bit too noisy, but maybe better when you're being charged at by a boar or aurochs.
To summarise, it would be a game changer in short engagements but not so in open field battles.
Finally, if none of these advantages are convincing, it does lead to significant reductions in training cost vs. a regular bow. The question is how maintenance and manufacturing costs would compare to the savings in training. My guess is that having a semi-proficient unit of IL users vs a semi-proficient unit of archers would be more cost-effective, but having a professionalised unit of IL units would not pay off.