does the US generally let firms have such leeway and lack of paperwork with hiring?

I don’t know for sure. With their at will employment, I would be surprised if they had such requirements in most states. In Czechia, we have no such requirements. We brainstormed interview questions in the office hours before the interview.

UK job applications have the requirements, essential and ideal, written out beforehand

Writing them ahead of time does not really change my point much. Write many requirements if you prefer hiring “on merit” or as few as possible if you want to give preference to diverse candidates.

Thank you for the time, effort, and thought out replies.

Thank you as well, it is so refreshing to be able to genuinely discuss and find common ground about topics like this these days.