Having recently covered this topic in a course for pre-service elementary school teachers, I thought I would write a bit about the somewhat subtle difficulties entailed in tackling this question. I am going to use language that may be at the level of undergraduate majors in mathematics or mathematics education, but I believe that the content can be scaled so that it can be implemented at the elementary level.
This answer is to some extent a re-hashing of other responses here; however, I am confident that multiple presentations of similar material can still be valuable in developing our own understandings.
As the OP writes:
I feel like the discussion ultimately breaks down into one in Group Theory with different sets and operators.
In a related spirit, I will list below four possible difficulties (phrased as questions).
When we use a binary operator (in this case, division) in elementary school, what are the associated sets under discussion (and what do we need to know about binary operators)?
If our ultimate goal (in elementary school mathematics) is to come to a consensus that $a\div0$ is undefined, then how will we explain what we mean by this word (undefined)?
If we are working with the rational numbers, then what background knowledge do students need?
What are the different interpretations of division, and how can we use them to make sense of an expression such as $a\div0$?
In thinking about 1, we might recall earlier discussions of the binary operator of addition and the set of whole numbers. In particular, we can take as input any two whole numbers (let us say order matters, though we would show at some point that addition is commutative), and give as output a unique whole number. The uniqueness here corresponds to the concept covered in an undergraduate mathematics course as well-definedness. For early mathematics education, even this feature is not obvious for addition: Consider an interpretation of addition that involves combining and counting discrete objects, and note that (cf. MESE 5866 on counting) "children sometimes believe the same collection can be characterized by two or more numbers; yes it has 14, and it also has 15!"
Subtraction presents problems of its own, because, unlike the above case, we may take as input two whole numbers in a particular order, but the output might not be a whole number. For example, if we are given the whole numbers $3$ and $4$, then we find that $3-4$ is not a whole number. How does one deal with such a scenario? The mathematical temptation might be to extend to the integers; early on in one's mathematics education, though, the simplest subtraction problems (i.e., those that involve an action: "take away/separate problems") are only posed if the minuend is greater than the subtrahend. We do not begin our discussion of subtraction by showing a child $3$ apples and asking that s/he take away $4$ apples.
These brief remarks about subtraction relate back to 2; namely, if we are talking about subtraction strictly using whole numbers, then an expression that cannot be evaluated as a whole number is said to be undefined. It is not that $3-4$ is secretly negative one; rather, we have not defined what happens when the minuend is less than the subtrahend, and so any such expression is (theretofore) literally undefined.
Briefly, with regard to 3: Doing justice to the (let us say positive) rational numbers is nontrivial, for it requires students to grasp the fractions (ordered pairs of a whole number and non-zero whole number) as well as their standard equivalence relation and the corresponding set of equivalence classes (cf. MESE 1447). Therefore, discussed below is only the case of why a whole number divided by zero is said to be undefined in elementary school mathematics; similar reasoning can be extended when one wishes to discuss more generally the division of rational numbers, but is not pursued here.
More precisely: Just as subtraction problems are initially presented only when the minuend is greater than the subtrahend, division problems for whole numbers are initially presented only when the remainder is zero. Sometimes a discussion of remainders is used as scaffolding in helping to deepen students' understanding of division; one must be careful, though, for the notation involved in this endeavor can be misleading. E.g., $4\div3 = 1R1$ and $3\div2 = 1R1$, but we would not wish to conclude that $4\div3 = 3\div2$.
As to 4, there are essentially three different interpretations of division. These are sometimes referred to as: the partitive (equal sharing), quotative (measurement AKA repeated subtraction), and missing factors interpretations. The former two interpretations are both discussed in JPB's answer; the missing factors interpretation states that $a\div b = c$ means there is a unique $c$ for which $a = b \times c$ (cf. my comment).
To answer the OP's question:
How do/would you explain why division by zero does not produce a result?
Recall that in the notation $a\div b = c$ we call $a$ the dividend, $b$ the divisor, and $c$ the quotient. In both the equal sharing and measurement interpretations, the dividend refers to the total amount of objects. In the equal sharing interpretation, the divisor is the number of (equal sized) groups of objects, and the quotient is the number of objects in each group. In the measurement interpretation, these meanings switch: The divisor is the number of objects in each group, and the quotient is the number of (equal sized) groups.
To resolve a question such as $a\div0$, it may be wise to begin with a few other questions:
Use each of the three interpretations to explain why $6\div2 = 3$. For the equal sharing and measurement interpretations, what might the corresponding pictures look like?
(Sketch: In equal sharing, we could begin by drawing two circles to represent the meaning of the divisor; then we would alternate putting one dot in each group until all six were used up. At the end of this process, there would be three dots in each group, which tells us that the quotient - i.e., the number of objects in each group - is three. Alternatively, using measurement, we could begin by drawing six dots to represent the dividend. Next, we note that the divisor tells us there are two dots in each group, so we begin to draw circles around the dots, two at a time. At the end of this process, there would be three groups of two dots, which tells us that the quotient - i.e., the number of equal sized groups - is three.)
Using the missing factors interpretation, $6\div2 = 3$ means that $6 = 2 \times 3$, which is a true number sentence. Moreover, we ought to observe that $3$ is the unique whole number that, when multiplied by $2$, gives $6$. A smaller number multiplied by $2$ will be less than $6$, and a bigger number multiplied by $2$ will be greater than $6$.
From here, we can segue into a discussion of $0\div6$ and $6\div0$; then, finally, $0\div0$.
I believe that you will find all three interpretations can be readily applied (as in the parenthetical example using $6\div2$ above) to each of the former two scenarios. The most difficult case to discuss is that of $0\div0$; again, pushing students to explain what is meant in each case (as suggested by JPB) will allow them to understand why dividing by zero (in the context of whole numbers) does not make sense.
In such a discussion, I would emphasize that the output is supposed to be a unique whole number. Without this attention to binary operators and underlying sets, one difficulty you might run into is that students believe, e.g., $6\div0$ is undefined (there is no whole number that, when multiplied by zero, gives six) whereas $0\div 0$ is "all numbers" (because any number, when multiplied by zero, gives zero). I believe that the latter remark ought not to be viewed simply as a misunderstanding, but rather as an authentic effort to make sense of the mathematics. However, it asserts that the answer is a(n infinite set) of whole numbers, whereas we have required (by definition) that any admissible answer be a single, unique whole number. Bearing this in mind, one should be able to convince elementary school students (better: have students convince themselves) why it does not make sense to divide a whole number by zero.