No, not really, it's basically incidental, and actually points to the fact that, pertinently, the word "supplier" has a strong connotation of referring to a business relationship.
Yes, a completely different word, in a now conpletely divorced and radically different context, doesn't work exactly the same as the original word we are discussing. Well done.
In your new example, the phrase "your driver" does not refer to a business relationship, so I guess saying "I'm not your driver" to someone because you don't want to ferry them around is now also poor use of language? Just stop dude.