I've studied physics in undergrad but I still made the mistake you describe in your last paragraph. The Grasshopper intuitively feels dynamically like an inverted pendulum but even harder because all you have are rockets at the bottom and you can adjust their rotation (I assume?).
On the other hand, there is no fixed pivot point around which you get torque, is this what the difference comes down to?
Do you happen to have any pointers to some articles that discuss this, and how difficult it is to balance? How does the wind affect things? What are the biggest challenges?
The inverted pendulum case (as approximated when balancing a pencil on its tip) can be reduced to a single object resting on the surface of a frictionless sphere, under the influence of gravity. At the precise top of the sphere, it can stay stationary. Perturb it even slightly, and it begins to accelerate away.
With the rocket, the object is essentially resting on a flat frictionless plane. If you tilt it a bit, then it begins to slide away, but the acceleration is always proportional to the initial perturbation. This is because the rocket always remains in the same orientation, and so the force always comes from the same direction. That, in turn, is because the rocket engine moves along with the rest of the rocket, rather than being a fixed point in space.
Think about it another way: without external support, and ignoring the influence of the atmosphere, a rocket is in free fall. The fact that it's accelerating to counter the force of gravity doesn't change that fact. Again ignoring the atmosphere, a rocket firing with 1g of thrust counteracting gravity behaves the same as a rocket firing with 1g of thrust in empty space. The second case is obviously not unstable (what would it be unstable with respect to?) and so the first case isn't either.
Adding the atmosphere back in, I could be way off here, but I think what little effect it would have at low speeds would be beneficial. Since the goal is not to move, the drag from the atmosphere will help make sure that happens. Since the rocket is more or less of a uniform shape, that drag shouldn't end up applying much of a torque either.
Edit: forgot to mention, I'm afraid I don't have pointers to articles. I've collected this stuff through various readings and thought over the years and don't have anything specific to point to. I think I first learned about Goddard's pendulum mistake in high school physics, for example.
Could be that the weight savings in fuel aren't enough for that choice, or that there are other good reasons not to (added complexity, potential non-reusable components)
[1] Although, aeronautics is famously non-intuitive, so ymmv.
Parachutes have lots of trouble. The rate of descent they achieve is still fairly fast, so you need something to brake the descent at the end.
This is why various American capsules splashed into the ocean, as well as the Shuttle boosters, because the water acts as a cushion. However, dunking your delicate rocket in salt water is a bad idea. The American capsules weren't reused, and the Shuttle boosters required enough refurbishment afterwards as to make it not really worth recovering them.
You can land on land, but then you need some seriously hefty landing gear, or retro-rockets. The Soviets use retro-rockets fired at the last moment to recover their capsules on land. A rocket has problems that a capsule does not, though, namely that you need to keep it from tipping over once it lands! You can make a capsule fat, but a rocket really needs to be tall and thin, and any sort of horizontal velocity will have it falling on its side. You also have very little control over where the rocket comes to a rest. Again, tolerable for a capsule, but dropping your rocket into a forest is not going to be conducive to reusability.
You can add wings, like the Shuttle did, and land horizontally. Trouble with that is that wings are extremely heavy. Each pound of wing in the Shuttle was a pound of payload it couldn't carry. If you try to fit every stage of a rocket with wings so it can come back and land, you'll end up not having much excess capacity left over to actually carry a payload. Wings are great for aircraft, when you can use them to efficiently fight gravity for the whole flight. But they suck for spacecraft, where they don't help at all on the way up, and are just dead weight for most of the flight.
So landing a rocket vertically on its engines isn't really all that bad in the end. Inelegant, perhaps, but it has a lot of advantages. You get absolute control over where the rocket lands. You can land it as softly as you like. You can ensure that it touches down with no horizontal velocity so as not to tip over. You use the same engines that you used for the launch, so there's no extra weight there. The only real downsides are the need for some sort of landing gear, and the need to carry a bit of extra fuel. Neither one is all that bad.
When considering the stability, don't make the common mistake of comparing it to balancing a pencil. When the rocket turns, the motors turn as well, so it is not inherently unstable the way you might think. Rather, the stability is neutral: if you perturb the rocket, it will just stay in its new position, not returning to how it was before, but neither does it run farther away. A rocket that's slightly tilted will stay slightly tilted (gradually accelerating sideways, of course), it won't tilt over further the way a pencil does.
Note that Robert Goddard made this same error. He built his first rocket with the engines at the top, believing this would help with stability. So if you were thinking that way, don't feel bad: the father of modern rocketry did the same thing!