When somebody asks for an order in writing, it's patently clear that the trust has already been broken, and irreversibly so. Nobody asks for that from a position of trust.
Unless it's multipart or otherwise complex, then the person asking just wants to make sure they didn't forget everything important. I've asked this on more than one occasion and I think it's clear that I'm asking so I don't forget anything (and have sometimes made this explicit).
No it's not. I made it clear to my last team that I'm just super forgetful and if you don't write down your request, I'll have to stop and write it down myself, implicitly making me an over-paid secretary.
That lack of trust is reasonable though - what’s in your head and what’s in my head may be completely different, why ‘trust’ that we’re on the same page when we can literally ensure it by writing it down and looking at it.
You wouldn’t verbally describe a structure to a builder, and then accuse them of not trusting you when they ask to see a blueprint.