So, polys _do_ sort of work, but not in the way they are presented to the public.
The rationale for continued use in clearance screenings is that it's not scientific, but it's better than nothing.
https://en.wikipedia.org/wiki/Polygraph#United_States
Interestingly (last paragraph of the link above) the US has indicted people for offering to teach other how to beat the test!?
Polygraphs as a power play - well, I'm sure it happens but that probably isn't the reason it is in the room. Maybe the polygraph is a ruse to make people think FBI questioners are stupider than they really are? I can see value in having a distraction from the actual questions.
Those are questions on the SF-86. Ditto for a lot of customs & immigration paperwork. "Idiot detector" questions.
But if you don't have them on there then they can claim "but you never asked".