Me feels like the usage of "inevitable" is to try to push the discussion toward religious and god or something like that.

Maybe not.

When we look at the evidence, it seems like life began pretty much as soon as it could have as the earth cooled. It could be that life starting on planets like ours basically just happens as a byproduct of the planet building.

Or, it means that the conditions under which life could arise are not persistent. Life either originated quickly or it would not originate at all. For example, if life originated in a small warm planetesimal, it only had maybe 10 million years before the short lived radioactivity in the early solar system (which we know was there from the decay products) died away and the planetesimals froze.

No, not at all. It was to try to push the discussion toward questioning our ability to predict the probability of life arising. The "random" argument is typically an argument from incredulity, "how could random processes make such a specific thing? The probability is astronomically small." But with only one universe observed, and only one instance of life observed within that one universe, we don't have any way to say that random processes just accidentally formed life vs the probability that it is likely, or even inevitable, that life would form given the properties of the universe.

Adding a god to the discussion doesn't really advance anything at all, honestly. It just explains the mystery with a bigger mystery, so it doesn't explain anything.

Given enough time, even with a random process you will have iterated through every possible result some day. So inevitable here means, it would have to happen at some point, simply because it is possible. So, math.

That highly probable events actually happen can be a reasonable practical assumption, but definitely not a logical certainty. It is the hard, probabilistic version of the sorites paradox: with few atoms and little time, any chemical process is unlikely; with a whole planet and a few billion years, chemical processes are very likely; at some intermediate sample size you can feel justified to assume that what you are interested in happens, but it is only a feeling, not a real qualitative threshold.