Yeah I had to laugh about that one as well. Especially the part about ImageNet being too general. If they knew anything about deep neural networks, they would understand why these base models are important and necessary for a candidate to understand, because they can be used to form a foundation for the solution.
In many cases, you can use the model as-is (and likely pre-trained) for a use case outside of "match label to an image", with only some additional training with your specific dataset - which may only be tangentially related (or not even that!).
These models (and the surrounding tools) have become for many problems more like Lego in my opinion - which is a good thing! It means they are more approachable for everyone, rather than being something mysterious and complex. Ok - if you dig, things become complicated quickly, but for many problems, you don't have to worry about these internals.
Are there problems which don't fit neatly into using a modified ImageNet or LeNet or one of the other "standards"? Certainly. But I think a candidate who understands the standards and basics well is likely a better one than one who only understands a specific subset for a particular industry (if there even is such a thing, which I am sceptical of).
Furthermore - it would be even better when a candidate can say "you know what - your problem doesn't need a neural network of any kind, let me introduce you to " - because there are tons of problems out there which can still benefit, and be a robust, easy to understand, and fast (to implement, to maintain, to execute - whatever).
I love neural networks, certainly - but there's been so much hype in the news, everyone thinks they need one, which will probably lead to many investing money into worthless (or expensive) solutions, where simpler (but less "sexy") ones would have sufficed.
The candidate who could know and tell the difference would be even more ideal - being able to interview/walkthru that might be a way to get a better candidate.