It's worth noting that the rights we have already impose responsibilities on others, whether they're negative or positive.

In political and moral philosophy, there are two different conceptions of "rights." A negative right, such as the right not to be harmed, imposes on others the responsibility not to do something to us. A positive right, on the other hand, imposes responsibilities on others to do something to us or for us. One such example of a positive right is the right to a lawyer; In the supreme court case known as Gideon Vs. Wainwright (1963), the Supreme Court ruled that the 14th amendment to the U.S. Constitution obligates the state and federal government to provide a defendant with a lawyer if they request one, free of charge. This is a quintessential example of a positive right in the same vein as the rights Roosevelt proposed here.

Roosevelt's proposal of a Second Bill of Rights is in fact consistent with the conception of rights as most moral philosophers, political philosophers, and legal experts think of it today. It's merely a list of positive rights that he thought we enjoyed in virtue of the fact that they were granted to us by the Constitution, or perhaps, because we are merely human beings.

One last thing: rights, whether positive or negative, always impose duties on other people to obey them. This is thought to be the case in deontological ethics, which was supported by the likes of Immanuel Kant, Thomas Nagel, and other moral and political philosophers.

tl; dr: Roosevelt's Second Bill of Rights isn't an aberration of the concept of "rights" in the views of many, if not most philosophers and legal experts.