I am not persuaded. Britannica says that a factorial is "the product of all positive integers less than or equal to a given positive integer [emphasis added]." Not a given "non-negative" integer. Thus, 0! is undefined.

Dividing 1! by 1 would compute 0! if there were a 0!, but there is no 0!. It would make as much sense then to say that -1! is 1/0, only, there is no such thing as 1/0, just as there is no such thing as -1! and no such thing as 0!.

How many ways can you order the empty set? It's not clear that it can be ordered at all, as it has no members. Another undefined notion, I suspect.

Meanwhile, I'm sure the gamma function is cool, but putting it in a "math made simple" article made me chuckle. I guess we each have our own notions of simplicity.