It isn't that a novel form of primitive life arises then gets eaten; the building-blocks necessary for a novel form of life to arise get eaten before those building-blocks have much chance of interact at all. From what we can tell it would probably take many millions of years of chemical interactions in the appropriate conditions after being seeded with carbon-based molecules from comet/space impacts (since carbon compounds seem to be relatively common).
It seems likely that life will spread and adapt somewhat quickly since it is usually easier to find new resources than enter an arms-race over existing resources. That leaves a very narrow window of time (in evolutionary terms) for life to arise >1 time independently then evolve enough to be on equal footing with the other form(s).
It also seems far more likely that if this very rare event takes place one of two things happen: 1. One form of life has a head-start in the arms race and overtakes the other, completely wiping it out 2. The two forms of life are compatible enough to exchange genetic material because they evolved from the same raw building blocks deposited on the planet; though it mostly fails to produce viable offspring in a few cases the forms successfully cross and the new third form of life successfully outcompetes its progenitor forms.
As for Mars, it seems likely that life never evolved there at all.
The next best explanation seems to be life never got beyond single-cell organisms and struggled to eek out a short existence. Before Mars dried out and/or the massive hemisphere-shattering impact struck and erased what traces of life once existed.
The simplest bacteria today are amazing organisms. They have cell membranes, which separate the protoplasm of the cell from the outside the world, and only selectively let some molecules through. They contain a blueprint to produce catalysts for a variety of chemical pathways; they can store energy in chemical form and use that energy to catalyze these reactions. The can break down complex molecules into simpler molecules and reuse these building blocks for their own purpose. They can move, to seek out new food sources. They can produce toxins that interfere with the chemical pathways of other cells, killing them.
But when we talk about abiogenesis, we aren't talking about the spontaneous generation of a cell. We're talking about the spontaneous generation of a single molecule, or a small set of molecules, probably proteins (amino acid chains) that, in the presence of proper raw material (amino acids) will tend to act as a catalyst for the formation of more of itself.
That's it; the reaction does not even have to be fast or obvious, even. If it takes years for the molecule to catalyze its replication, that's fine. Everything on top of that comes down to evolution, the slow development of a new trait - cell ways, a genome separate from the cell machinery, the ability to metabolize organic molecules into raw materials, locomotion - over millions of years.
So in addition to xenadu's point, there's also this: the abiogenesis of new self-replicating molecules could still be happening today, and we'd probably never know. The objects in question are too small, and the time scales involved are too large.
Since the primordials soup times, we also have energy hitting inanimate matter. On this planet. And also we have the other planets in our solar system. Why aren't we finding lots of bacteria being spontaneously generated everywhere?