Are you not aware that most languages provide that for free? Note that the question was about implementing, not using. People use them all the time, but they don't reinvent the wheel to do so when it is in the standard library and has nice abstractions already.
Why is a web developer implementing a DFS by hand every day? I would not hire that person.
I'm not sure what you mean. I don't know any languages where you don't have to define your own node if you want to implement a new kind of tree, and write a bit of code to recurse over it. (In JavaScript the most common tree is the DOM, but you should still be able to build another one if you need it.)
Also, in an interview it's perfectly fine to say you'd use a library, if you know one that applies. (They might ask you next how you'd do it without a library, but you still get points for knowing about it.)
The other thing to remember is that everyone is effectively grading on a curve, and with practice you get better at interviews.
Why is a web developer implementing a DFS by hand every day? I would not hire that person.