Wow...I am flabbergasted. My adult daughter has been on my insurance (and will remain as long as I can legally keep her there, as it is one "little" thing I can do for her) and I have never received a bill or a call asking for payment on her behalf. All I get are monthly statements from my insurance company that shows what doctor she visited, when, how much was billed, how much insurance paid, and how much the patient "may owe." I never actually receive anything from any doctor's offices or actual bills - they all go to her in her name.
When she had her major accident and we had to sign for procedures, her case worker was the one who told us that she was financially responsible for all care because she was over 18, even though she was on our policy. It was a huge relief as her cost of care would have bankrupted us. We are still financially digging out of what it cost from loss of work, lodging, food, etc., as she was sent quite far from us and was in such bad physical and psychological states that we could not leave her for quite some time.
I would in NO WAY think it makes you "cold" to worry about who is financially responsible. Whether our adult children are completely unaware of the financial repercussions or truly just don't care, they would strip us of everything we spend our lives working for if we don't at some point say "enough."