Re: "if a user sees the business's name then they are more likely to actually take the call"
"In in US the hospital gets into privacy problems if they undertake to call you about a relative you do not have Power of Attorney in healthcare matters for."
But they're also legally obligated to notify Next of Kin, especially in a morbid or mortality situation. Most hospitals I know leave a caller ID, but it's usually just the "front desk" number, not any specific ward or clinic, to at least inform the caller it's a hospital.