It is absolutely trivial.
>"but you'll also get tons of interview questions about skills and projects and challenges from your current job"
Your last job unless it is 10 years old should provide all of the answers. Any really identifying details should not be asked / answered as the employee is normally under NDA.
Same for references. Reference from "current job" can simply be refused. I can hardly imagine employee going to their boss and asking for a job reference while still working.
Anyways I am independent and run my own company. Maybe I am not up to date about how deep the US employers are able to stick their fingers up that proverbial hole.
These fall perfectly into experience with last job. Does not have to be current. And all those questions you asked are trivial. Also I've never dealt with the recruiters. I have always searched and found perspective companies myself and no they were not Amazon big type. If I could not speak with the owner I would simply walk away - not my kind of place.
My first programming job in Canada - I just simply walked into the office and asked to speak to the owner (I knew it was small 20 person consultancy).
Since 2000 I am on my own but I still find clients and have interviews. Just a different type of interview of course.
Also in this case I really don't mind disclosing - I was laid off because my position was made redundant entirely.