The elderly, poverty stricken, disabled, infirmed, widows and orphans, disabled soldiers and first responders all are "entitled" to societal support to one degree or another. Those are the norms of our society.
To deny this is to deny the social contract and civility. No society can exist without them. Even wolf packs have norms of conduct.
Rand's point goes beyond this. It's a much more concerned with individual senses of entitlement. You, DrJBHL, are very successful because you work hard and apply yourself and I, Kantok, am not, because I'm lazy and want to spend my days reading romance novels. Because of these facts you owe me. The fact is that I'm not one of those special classes of people. I'm a healthy 30-something who could work at least a minimum wage job, but I just don't want to. I want to spend my time reading Danielle Steele. I am still entitled to your charity, solely because you are successful.
And it goes further. I can actively disparage you for being greedy, evil, and cruel, despite the fact that you are successful, treat your employees well, and are otherwise a generally charitable person towards me and society as a whole. Despite this disparagement YOU STILL OWE ME. I am still entitled to the reward of your toil and your mind.
In fact it goes even further. I can actively work to undermine your success and yet you still OWE ME.
The level of entitlement Rand is talking about isn't the civil safety net. It's not in regards to caring for those who have otherwise contributed to society or cared for themselves but no longer have that capability. It's about the people who are perpetual takers BY CHOICE and who believe they have the right to take from the successful by virtue,and this is important, of the fact that others are successful and they are not. Nothing more. The circumstances of your success and the circumstances of my failure are irrelevant as is my behavior towards you.
You have and I don't and therefore I am entitled to what is yours.
People have used Rand's ideas to argue against the civil safety net like you mention, but her argument is against a much baser human emotion. Jealousy of and theft from those who find success solely because they find success, writ large and given legitimacy by the support of the masses.