That's a nice thought (and I do "like" that you shared it). But in communication, what the other person picks up is what you've communicated. And when people are in pain or struggling with other real-world difficulties, what they're likely to pick up from these cards, or similar displays from strangers, is not "they are a significant part of something bigger." It's more like "Actually I think you look pathetic. I don't want to know."
"You are a significant part of something bigger" is not part of everyone's belief system. It would never cross a lot of people's minds. Until I read your explanation, it's very unlikely that that interpretation would ever have crossed my mind, even though it does relate vaguely to one of the more controversial and obscure Christian beliefs.
In the absence of that belief system you're talking about, a card printed with "YOU MATTER" poked at a person by a stranger might communicate:
(1) "You look pathetic. I don't even want to know, because it's probably contagious, but your pathetic condition bothers me."
(2) "I am advertising something you probably can't afford AND don't want."
Or, at best, (3) "I found this card. I don't know how to read. Please can you read or translate it for me?"
I'd find any of those messages annoying; a person whose pain included depression and self-blame would probably default to #1 and might go straight home to commit suicide.