Re: "if a user sees the business's name then they are more likely to actually take the call"
Not even a hospital you've never visited, but your dear relative ended up in their Emergency Room?
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. And how would said hospital know of that? I have a mobile that has ICE ( in case of emergency) numbers set up. The medical staff ( first responders included) know how to access that so the call would come from my phone not the hospital's or EMT's. Phone destroyed in said emergency and we are all SOL.