Is the Drake equation based on a logical fallacy?
To directly answer the question, the Drake Equation is not based on a "logical fallacy" (which means a different thing, philosophically). The logic is fine, the real problem is we don't have any meaningful data.

Regarding abiogenesis (the first formation of some simple life): as you correctly state, we know only that it happened once (here, presumably but that doesn't matter). However the argument that it happened quite quickly on our planet is meaningless statistically, because that's just a sample of one.

(If anyone thinks this answer is too waffly or off topic just say so and I'll delete it by the way; I think the question is a fair one but we just can't present any data, just reason.)