Yep, a nonsense argument. It is not because something has started only recently that it will end soon, or vice versa. You could just as well argue that the human race has an inherent life span of 1 quintillion people, and that because we are only at number 12 billion now, the impending dommsday is still far far away. This argument is equally rubbish.
A better way would be to calculate the average lifespan of a species on Earth that has survuved for at least 100,000 years (I mean from evolution till extinction, not the lifespan of individuals), and compare that to ours. It would still be only a statistic, not a certainty by far, but it might give a better idea.

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Knowledge is a curse, but ignorance is worse