RoboJesus Of course they can, there are so many ways to get an egg from the first floor to the ground, you can add all sorts of different constraints, you can test and evaluate different materials, it is a very open-ended engineering problem.
It is also the kind of simplistic tick box engineering problem that is easily solved if you have given constraints, but once you've solved it through simple padding, and calculated the forces involved, you can change the materials and use that to find out information about the materials. Or you can switch to destructive vs non-destructive devices - ie padding generally only drops one egg, we want a re-usable device. Or you can switch to slowing the descent rate with parachutes and how much you need to slow it by, what is the terminal velocity of the contraption etc.
A KS4 student is not ready to make interesting discoveries about String Theory, so whilst they can read about it, they're just learning from others. A much more valuable skill is learning to solve your own things, because eventually the genius is going to run into problems others haven't already solved, which is why JustRichmal's maths is about solving it rather than reading about other solutions.
If your child has already done JustRichmal's things just have them do whatever they want to do next, just carry on teaching themselves maths and stop pushing problems on to them, they are already at the level where they need to learn themselves and not be fed.