I disagree. Arguments to symmetry like this are found all over the place, especially in the kind of geometry and physics I'm familiar with. In fact the solution I came up with when I read the problem was very similar.
You're right that it's not a proper explanation. This feels just like a case of "details are left as an exercise". This is most frustrating when you're learning a subject, and you want to be able to check the details, or you are skeptical of a claim. Inevitably though people elide things that they think the reader (who they model as similar to themselves) will immediately be able to guess based on their experience. I'm sure you can think of an equivalent situation in your field of expertise.
I can think of things where I would leave out details that a non-expert would need filled in. Except, I wouldn't do that in an interview situation where someone is specifically asking me to explain that one thing. (Yes they started with the giant coin thing, but "symmetry" is still the substantial majority of their argument.)
The sibling to my first comment says it a lot better than I did: in a similar but different situation there's symmetry but no analogous strategy, so further explanation really is needed here.
In a sense, that makes it even more damning. Eliding details that you don't realise are important, but actually are, shows a real lack of understanding of your own argument. And that's something I have definitely witnessed in my own field!