What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?

I think of the requirement $0\neq1$ for fields as a consequence of the desires for (1) fields to be integral domains and (2) integral domains to satisfy $0\neq1$. Of course that just shifts the question to: Why do I want integral domains to have $0\neq1$? That desire comes from my (quite general) inclination to treat the empty set just like other finite sets. Here's how that's relevant: The key clause in the definition of integral domains (besides the axioms for commutative rings with unit) is that if $xy=0$ then $x=0$ or $y=0$. It follows immediately by induction that, if a product of $n\geq 2$ factors is $0$, then there must be a $0$ among those factors. The same holds trivially for $n=1$ under the obvious convention that the product of a single factor is that factor. So I'd like it to hold also for $n=0$. Now the product of no factors is $1$, so what I want is that, if $1=0$, then there is a $0$ in the (empty) set of factors. As there is no $0$ (or anything else) in the empty set, I infer that $1\neq0$.

(The same underlying idea explains why I don't regard the integer $1$ as prime and why I want lattices to have top and bottom elements.)

And Bourbaki said (Algebra I, Chapter 1, §9, definition 1)

"A ring $K$ is called a field if it does not consist only of $0$ and every non-zero element of $K$ is invertible"

and fields did not consist only of $0$.

Bourbaki saw that the rings were good, and he separated the fields from the other rings.
He called the fields with commutative multiplication "commutative fields" and non-commutative fields he called "skew fields" .
And there were domains and there were division rings-the first day.

Edit: For miscreants, heretics, apostates, schismatics, infidels and other iconoclasts
Since your ilk might not know: this answer was shamefully plagiarized from lines 3,4,5 of this text.

The zero ring not only cannot be a field, it cannot even be either a local ring or an integral domain. If it were either of those things, the ideal $R\subset R$ would need to be prime for any unital commutative ring $R$, which is awkward and pointless, creating unnecessary difficulty everywhere.

This is pretty much the same reason that $1$ is not considered a prime number.

There are hypothetical advantages to a field that includes into every other field, but these are not realized in practice by the zero ring. In a typical examination of the "Field with one element", one looks at something that isn't a ring at all, such as the monoid $(\{0,1\},\times)$.