The response to your question has several moving parts. The philosophic positions of functionalism and stratification are only possible in their extreme when viewed from a theoretical perspective. Stratification, by its most general description, is classifying people concerning their social, economic, education, or racial (among many others) status. Functionalism is a form of categorizing people as well. In place of the stratifying class descriptors, functionalism recognizes the complexities of society. Institutions form the framework in which norms, customs, and traditions (among others) play out daily. Cleanly demarking the boundary between the two positions in practical terms is not an easy task!
Humans do not live in the abstract but rather the concrete. In a utopia, one theory conceivably has more appeal than the other. The profound question is, which one? Few people will disagree with the notion of ending the inequities found in a global society, which are the result of stratification or fundamentalism. It is hard to imagine any human who is not emotionally moved by famine, poverty, mass immigration (which is the result of conflict), or the myriad of other social plagues. However, neither position offers much in the way of hope.
The social structure of the world will not blend into a homogenous global community where the common good transcends the human condition of trying to better one's position over another. To repeat an often-heard phrase, "the heart of the problem is the problem of the heart." Theologians, philosophers, politicians, social scientists, and anyone involved in the work of humanists since the first day humans appeared on Earth have been confronted with the moral dilemma of social inequality. Valiant efforts addressing social inequalities are evidence that humans desire to change, but the complexities of culture, religion, history, politics, personal ambition, and worldviews stymie large-scale good works.
A world dominated by either idea is not a world where most people will want to live. The notion of a classless society, while admirable, is an unlikely possibility. Inequitable distribution of resources is a fundamental fact of the human condition. Humans are, by nature, not always rational, logical, or philanthropic in placing others above their own self-interests.