You're mistaken. Life thrives in a huge variety of environments and survives in many more, and we have evidence of similar suns and the existence of planets in habitable zones. We can observe varieties of axial tilt and configurations of moons in our own solar system. We can form (bad) estimates of probability for earth-like cosmic conditions, and they're good enough that we've found some planets where we expect they might be.

It seems like a far larger assumption to think Earth is unique in the cosmos, given nature's tendency to repeat itself. There are numerous natural harbors on earth, for example - places where the sea currents and land terrain combine to make a relatively calm environment even during stormy conditions. If we view Earth and the solar system as a natural harbor in space, what basis is there for thinking it can not exist anywhere else? The fact that we're uncertain about exactly how common it is is not a good reason to think the answer is 'not at all'.

Your first paragraph there is about how life spreads around on a planet once it's arisen. It doesn't have anything to do with the chance of life arising in the first place. We have evidence of precisely one origin of life event -- all life on Earth looks to derive from a common ancestor (called "LUCA").

We don't know how common OoL is on a planet like Earth. We cannot use the fact it occurred here to conclude it's likely, since if it didn't occur here, we wouldn't be here. Our mere presence to ask the question biases the observation. The less likely it was, the more biased the observation.

Second paragraph: what is this supposed tendency of nature not to repeat itself? The universe is full of objects that are unique arrangements of atoms, not repeated anywhere else. Indeed, all but extremely small objects are very likely to not recur. So this supposed universal property of repetition doesn't appear very universal. Proposing that Origin of Life is in the subset of phenomena that do repeat is just begging the question.

As for "what basis is there for thinking it can not exist anywhere else?", that question is relevant only if I were asking you to believe life is not common, not if I were asking for justification of statements that it is common. The opposite of belief in X is not belief in (not X)!

It's not assuming the conclusion. It's stating a fact that the universe is really big, and there's no kind of indication that the earth is particularly special.

It also just seems reasonable that, given the size of the universe, the odds of life forming exactly once seems much less likely than the odds of life forming more than once.

The assumption is in the "not so special" characterization of Earth.

"Just seems reasonable" isn't an argument.

Of course it is an assumption, but it seems to be a reasonable assumption unless there is specific evidence to the contrary. Much of our understanding of the universe is based on the assumption that we don't occupy a particularly special place in it.

I've heard theories about either the presence of Jupiter, or something about the nature of our moon as providing a safe long term environment for life to form on earth, but even that doesn't feel that special, when there are 10^21 stars with planets in the universe.

Now you are elevating life is common to a default position, simply by declaring it so. I disagree. Me, the complexity of the simplest living thing makes the default assumption that OoL is incredibly unlikely.

You should be aware that the logic you are using, vulgar Copernicanism, was used in the 1700s to conclude that the other planets in our solar system must not only have life, but intelligent life!

What your argument ignores is that we don't live on a randomly selected planet in the universe. We necessarily live on a planet that has life. This "observer selection bias" has to be taken into account; the rarer life is, the more biased our local observations will be.