It’s true that Rand made many expressions of his opposition to race discrimination in what was a hearty effort to blunt the effect of what he was saying, but it is not true that his “reservations” were limited to federalism concerns. (As to federalism, there was an argument, rejected long ago by the Supreme Court, that the Constitution did not empower Congress to regulate in this area.)
Rand was also expressing the view that owners of private businesses have a right to decide whom they will serve. Such a right would not run counter to the 14th Amendment, because the 14th Amendment only protects individuals from the actions of the state and privately owned restaurants and hotels are not the state. If you want a legal requirement that these businesses treat people equally, you need to pass a statute, which is why the Civil Rights Act of 1964 was passed. And that statute was susceptible to arguments it violated the right of the business owners to do what they wanted with their own property. When the Supreme Court upheld the Civil Rights Act of 1964, not only did it need to find an enumerated power for Congress to act, but it also had to deal with the argument that the Act violated the Due Process Clause. Rand’s statement harkened back to both of those old arguments.