Stopping evolution, or there will be more than three!
There are several issues with such an ecology. So for a moment we'll assume this is engineered somehow for that specific effect. This eliminates the need to explain how the ecology evolved to that particular state.
Given that we assume life to rely on some information-carrier substrate to direct the development of new organisms (DNA), and given how in all foreseeable circumstances mechanisms like that must (on a geological timescale) occasionally fail, it's difficult to imagine how evolution has been arrested. Any malfunction that causes a variant of an organism to appear (a mutant) carries with it the chance that this mutation will be advantageous. More such mutants should show up in subsequent generations.
Now, if any of these organisms are distributed worldwide (or even continent-wide), then there is enough range that the organisms are isolated from each other. They should (over a geological timescale) start to diverge into separate species.
Thus, if we're picking some arbitrary time period in the existence of this world, there's no reason to expect it to have only 3 extant species. You end up needing to visit it quite soon after whatever it was that caused there to be only 3 in the first place (or there won't be just the 3 anymore).
If you want this to not be true, you have to come up with a mechanism that arrests evolution, and I struggle to imagine anything that could do that.
Next, let's talk about how 3 such organisms could even comprise a viable ecology. It's true that you can find experiments (and I suppose, works of art) where someone has sealed up a big bottle or perhaps a sphere of glass with some plant in it, and some other animal (snails, perhaps, or shrimp, etc) and that these little ecosystems can chug along for decades (or longer, no one knows if there are limits with any certainty).
Your world has several advantages over these. In the "bottle garden" (that's what Wikipedia is calling them it seems), if the plant dies for whatever reason, it is either the only plant, or one of a very small number of them. This is disastrous. The dead plant won't magically resurrect, and even if it isn't the only one it can wreck the homeostasis of the bottle, causing the other plants to die. Generally, the animal life is a bit better off, you'll have more individuals and they are fast-breeding and perhaps even hermaphroditic/parthenogenic... a single individual dying is much less likely to cause a collapse. But with a small number of each, a death of any plant or animal is just a risk that doomsday is in the cards for this tiny little microcosm.
Your world has no such problem. Even if something sparked a local die-off, the planet is large enough that there is a reservoir of organisms to re-colonize that desert and restore some balance.
But we can't discount how precarious that balance is. The bottle garden doesn't suffer just from a shallow bench, so to speak... there's only three positions sitting on that bench. Each extra species (that fills a new niche) extant on this world brings a little more stability to it. Each extra species adds a slightly different nutrient cycle to the world, each extra species can be provide a backup nutrient cycle if some other is interrupted.
These overlapping cycles don't only substitute for others when disaster strikes, they more finely tune homeostasis. With many animal species, you don't have to worry so much about plants going nuts and over-oxygenating the atmosphere (they tend to look at the plants as food and some minor boom-bust cycle is about to occur). With many plant species, you don't have to worry about some overly-gluttonous animal deforesting the entire world. In the bottle garden, the physical size of the environment itself limits those behaviors... but you've given them an entire planet.
Finding the right balance between two species in a bottle garden is difficult and requires experimentation, skill, and more than a little intuition. Finding the balance on your planet-sized bottle garden will be even more challenging, though there's nothing to suggest that it is outright impossible.
Can any of this even work without decomposition?
The bottle gardens might only include a single plant species, and a single animal species... but that's far from the same thing as including only two species. Whether deliberate or unintentional, whether they were aware of it or not, those who have made these little tiny closed ecosystems were also putting bacteria (and possibly other microbes) into the bottle.
Thus, when one of the little fairy shrimp die, or when a leaf falls to the bottom, those get to work and turn the detritus back into something that the remaining plants can absorb as fertilizer.
Your world needs something like that too, and it's non-negotiable. Now, maybe they weren't included in your list because they're boring or it was an oversight, but you'll need to keep this in mind as you explore the idea.
What could even lead to just three species?
From what little we can speculate on the subject, it seems that at one time in our own planet's history there was probably only a single species. It was unicellular and microscopic. And unlike modern microbes, it may not have reproduced quickly... it might not have been "very competent" in that job (after all, it would only have to succeed once).
But after that, evolution kicked in, and it almost certainly diverged into multiple subspecies quite quickly. Whether this was hundreds, thousands, or tens of thousands of generations, this is still only a difference of a couple magnitudes of order, and whether or not a generation was measured in 20 minutes or 20 days... well, it just didn't take very long for there to be more than one type of little microscopic squiggly thing.
And then for those to proceed up through to the sorts of multicellular life we know today. To have the divergence of some photosynthesizing plants and scuttling arthropods, a skulking vertebrate... well, for that to have happened, evolution had to produce more than just three models.
Even if you could imagine a disaster that wiped out all but 3 of those, how likely is it that the 3 remaining would constitute a viable ecology by themselves? Why just 1 plant instead of 3 plants and no animals? Why only 2 animals and 1 plant instead of 3 animals and no plants? Why 1 herbivore and 1 insectivore instead of 2 carnivores?
The only way to reconcile this requirement is to suppose that the arrangement is artificial. Someone engineered this planet to be this way. Maybe something that looked like a terraforming effort, maybe K3 quasi-gods just wishing it into existence in the same way we'd run a simulation. But no natural set of circumstances could produce the result you're wanting.