By gear box do you mean the covered system between the bottom bracket/cranks and the rear wheel? Dutch commuter bikes are usually one speed and that system keeps grime out of the chain and keeps your pants clean. It is indeed a PITA to remove the rear wheel on these bikes (especially the first time), although you can often patch a tube in situ without removing it completely.
This product does not replace that system. It is for very high-end bikes with owners. These bikes still have a chain that runs front to rear, it only replaces the derailleurs. If it fails or the battery dies, you're stuck in one gear but you don't have to push your bike. And you can manually change gears if you really need to. Yes it has a battery, yes it's waterproof.
https://youtu.be/yKQBR1ohtIA I mean this. Whatever's the proper name for it. The video makes it look easy, but the reality of it is that
* Spokes don't really fit into the hole (and you aren't really going to pull a spoke out of the wheel to take the wheel off anyways).
* Adjusting the tension on the cable takes many tries. And you can't really adjust it without taking the wheel off. So, you put it together, try riding it and see if the speeds switch properly. Once you see they don't, you take the wheel off, pull the cable a little bit. Rinse and repeat.
Not to mention that the parts are very small and easy to lose if you have to make an emergency repair. It's very sensitive to dirt.
It looks more "compact" and out of sight, and from what I can tell from talking to my Dutch friends / acquaintances: punctures don't happen that often because roads are kept relatively clean. And people would generally just go to a bike shop and pay to have things fixed. So, decreased reliability and difficulty in maintenance don't bother them.
That's an "internal gear hub". It also takes some practice to do it quickly. In the video a small hex key is used to rotate it, not a spoke. You shouldn't need to adjust tension after re-install more than a couple of clicks on the barrel adjuster. If you do need to adjust at the hub, it does take practice which you normally won't get doing it only once per year..
They have a few advantages (basically zero maintenance over the life of a bike, can shift when stopped, easy to fully enclose entire drivetrain) and disadvantages (heavy, expensive, less efficient, can't shift under load, pain to remove wheel) compared to derailleurs.
They're nice for commuter bikes because you can fully enclose the drivetrain, keeping rain off the drivetrain and grease off of your pants and never need to be touched, until you have to remove the wheel, as you found out. Overall I think they make sense for that kind of bike.
This product does not replace that system. It is for very high-end bikes with owners. These bikes still have a chain that runs front to rear, it only replaces the derailleurs. If it fails or the battery dies, you're stuck in one gear but you don't have to push your bike. And you can manually change gears if you really need to. Yes it has a battery, yes it's waterproof.