They are absolutely forbidden to refuse emergency care to stabilize, and forbidden to turn away a woman in labor. By law. No matter how private they are, if they are open they must provide care or arrange for it to be provided.
Similar regulations cover many private businesses - only some of which are utilities.
Use whatever word you want, but Cloudflare is not regulated in the same way that a hospital is, and hospitals are not only regulated around who they can refuse service to. Regulations extend to the way that they advertise, which insurance they have to accept, what standards they have to follow in providing service, etc...
Banks are private companies too and they're regulated in regards to who they can turn away as a client. I would hazard a guess that Cloudflare isn't eager to be regulated in the same way that a bank is.
Keep in mind that for both hospitals and banks, the restrictions about what customers they can turn down are not as far-reaching as the 1st Amendment and public service restrictions are. What people are asking for with Cloudflare (don't take down a site's protection unless a court tells you to) is even stricter regulation than what a bank or a hospital has.
Can you think of any private entity where they're not allowed to turn down service to any customer unless they're first ordered to by a judge?
They are absolutely forbidden to refuse emergency care to stabilize, and forbidden to turn away a woman in labor. By law. No matter how private they are, if they are open they must provide care or arrange for it to be provided.
Similar regulations cover many private businesses - only some of which are utilities.