Thank you. Wikipedia agrees with you, and not with Britannica, as to "non-negative" vs. "positive." Other sources stick with positive numbers only.

I am inclined to value 0! at 1 by convention to avoid using of the gamma function when calculating nCn.

The idea that the empty set can be "arranged" in one way suggests that 0C1 is a thing and that:

0!/(0-1)!0!=1

i.e., -1!=1. But -1! is undefined.

Would it be better to say that nCr is properly calculated using the gamma function, for which, in the case of positive integers, factorial multiplication is a usable algorithm but (i) in other cases requires the more complicated procedure outlined in your article, and (ii) there is only one "other case," so why not use a convention?