I think you're taking her term "Rational Animal" and applying your own definition to it. Being a rational animal doesn't mean "Rational all the time." It means that, in order to survive, man must be rational. The extent to which one survives and is happy is the extent to which that individual is rational, even if it means only rational now and then.

Logic and reasoning are more recent inventions and used even less frequently.

Logic is as old as advanced civilization. This is not a coincidence.

I actually think civilization works better than individuals.

It seems like you're also misinterpreting Rand's message here. She doesn't advocate the destruction of society and civilization, and she is not an anarchist. Indeed, she speaks volumes of the value of society to the individual, and she has multiple essays on the role of government. And not that government is a "necessary evil," but that a proper rights-respecting, rights-protecting government is good because it is of value to the individual.

The collectivism (with respect to government) that she opposed is that of the majority voting away the rights (including property rights) of others. Democracy is two wolves and a sheep voting on what's for lunch.