"The problem, however, is whether you like it or not, you’re still conveying the message that your screen at that moment is more important than the person you’re with"

I think this depends on the specific situation. If I'm in the middle of a sentence and someone breaks their attention to check on the noise coming from their pocket, that's going to be a problem with me. However if there is a lull in the conversation, and we're both sort of sitting around enjoying some wine, go ahead and do a phone check. I don't need 100% attention just because we're in the same room. If you have to ask permission to check your phone just because you're in the same room, do you also have to ask permission to leave the room? What about picking up a book and reading? Does that require a permission check?

I'm not sure why there would be a particular bias against screens. If you were at a drafting table working and your child is staring at you, I see nothing wrong with ignoring the child and continuing to work. Screen or no screen. Most of the time the child's immediate situation is not the most important thing in the world. She's just fine, playing and observing. I don't think we do anybody any favours by guilt tripping ourselves just because we're busy and it's rather unrealistic to have the child expect full attention from all who surrounds them.