Most birds roost in trees to minimise exposure to predators. Is it possible that birds that are used to living with humans might similarly see a cage as a place of safety? For rest or just taking a break from watchfulness?

(I'm personally uncomfortable with birds being caged for long periods or in confined spaces, and I'm not offering the above as a justification. I don't own or live with any animals.)

Not OP but of some bird owners I've see that let their birds hang out in their house / on their shoulders and such the birds willingly go to their cage to rest.

By watching them, and advice from experts.

How did you arrive at the conclusion that they don't?