I've never seen such sentiment outside of joking around, but even accepting your claim, who cares? What difference does it make?

Because websites et al fail a lot and a lot of software that runs invisibly does not. Typically engineering refers to something that works at a particular rate of failure (low hopefully).

Okay, sure, you alluded to that earlier... But who cares? Even if what you say is true, that does not give it significance. You somehow completely missed the question. Let me try again:

Who cares? What difference does it make?

You have: "full stack" developers