Reading about the United Arab Republic, the dysfunction of the integration between just Syria and Egypt and the conflicts Nasser had with Iraq’s Qasim, was there ever a real chance for one nation to unite the Arabs?
Pardon my assumption, but I think you are asking two different questions here. While the success of Pan-Arabism was irrevocably tied to Nasser, I do not think the inverse was true.
If you are just interested in Nasser and his chances of success, I will point you to two of my other answers on Nasser.
https://www.reddit.com/r/AskHistorians/comments/ehruo1/what_was_egyptian_life_like_under_nasser/
Just to add a little to those two posts, I think Nasser was a qualified personal success. He did not achieve close to everything he wanted, but his impact on Egypt is still felt today. He is arguably the most important leader in Egypt’s modern history and he lives on as a myth and legend there and many other parts of the world. I cannot ever claim to know what was going on in his head, but he surely would have been pleased to know the renown and reverence he would receive.
As for Pan-Arabism and the possibility of its success, I think this passage from R. Stephen Humpreys in Between Memory and Desire: The Middle East in a Troubled Age sums up one of the major internal issues with the movement:
“Arab nationalist thinkers … had looked at the crucial problem confronting them and their people as one of identity rather than as one of institutions. The question was, who is an Arab, not how can the Arabs build a common political life and effective institutions of government? … Very few writers asked seriously how [the projected Arab] state would be constituted, how the relationships among its many disparate regions were to be defined, and how different social groups would be represented within the political system.” (66-67)
If you look at the major writers (Sati al-Husri, Michel Aflaq) and leaders like Nasser, they talked a great deal about what united the people of the region, the things they could accomplish when the Arabs were united, the problems with disunity, but were scant on the details of how a united Arab state would work. This would have a real limit effect on the movement. Irreconcilable differences over matters of rulership were one of the reasons the UAR (the United Arab Republic, the union between Egypt and Syria) fell apart so quickly.
Adeed Dawisha in Arab Nationalism in the Twentieth Century: From Triumph to Despair believes that a commitment to democracy, and an avoidance of authoritarianism, would have addressed these underlying issues, but that is attributing way too much power to a type of governance. In my opinion, no one addressed this issue because no one had the answer. How could they? The one modern model of holding these entities together, the Ottoman Empire, had just fallen apart and it also was what these thinkers were reacting against.
It brings up a philosophical question about whether an entity that large could exits in a world where states seem to fracture rather than cohere. But that is for a different post and a different historian.
So that is one black mark, perhaps the biggest, against Pan-Arabism ever working.
Another important one is apparent in your question. Dawisha rightly states that Pan-Arabism, or Arab Nationalism, was at its strongest in the 1950s and 1960s. That was also, uncoincidentally, when Nasser was its greatest champion. The strong linkage to the Egyptian president was a significant boon to the movement for his charisma, his credentials as a fighter against imperialism, and his international renown as one of the leaders of the non-aligned movement. Unfortunately, the linkage also meant that the movement lived and died with him.
Under his leadership you had the creation of the UAR, a short-lived but real step towards a Pan-Arab state. However, its impact and the impact of Pan-Arabism was limited by its association with him. With no clearly defined plan for how things would run, it was rightly assumed that Nasser would become the authoritarian head of any state, like he would in the UAR. This obviously did not sit well with leaders of other Arab states and mitigated any real movement drawing the states together. For example, anti-Nasser sentiment prevented Iraq from joining the UAR. The fear of being subsumed into Nasser’s Arab republic also brought real hostility to relations between groups that were meant to be part of an Arab whole. For example a proxy between Egypt and Saudi Arabia erupted in Yemen, something Nasser called “his Vietnam,” over their struggle for leadership in the region. The strong ties to the cult of personality of Nasser also meant that when it took its fatal hit, the embarrassing defeat in the 1967 war, it also struck a mortal blow to Pan-Arabism. In less than a decade Fouad Ajami would declare the movement dead. Whether that is actually the case is up for debate.
That brings us to a third roadblock to the success of Pan-Arabism. The fact that a Pan-Arab state would be a serious threat to the political desires of the United States and the Soviet Union for the region. This is getting into counterfactual history here because there was no need to step in to prevent an Arab union, but even if the Arab states had decided on a fair way of ruling a Pan-Arab state and gotten over the hangups of Nasser ruling that party and not suffered an embarrassing defeat in 1967, there is no scenario where the two superpowers would have not intervened to undermine it. Especially in the case of the US with its strategic interests in Israel and its distaste for anything related to communism, they obviously had no love lost for “Arab Socialism” (the imagined economic plan of Pan-Arabism).
So I think we can say "no," there was no realistic chance of Pan-Arabism working.
Sources
Ajami, Fouad, “The End of Pan-Arabism,” Foreign Affairs, vol. 57, no. 2 (Winter 1978 / 79).
Batatu, Hanna, The Old Social Classes and the Revolutionary Movements of Iraq: A Study of Iraq’s Old Landed and Commercial Classes and of Its Communists, Ba‘thists, and Free Officers (Princeton, New Jersey: Princeton University Press, 1978).
Cleveland, William L., A History of the Modern Middle East, 2nd ed. (Boulder, Colorado: Westview Press, 2000).
———, The Making of an Arab Nationalist: Ottomanism and Arabism in the Life andThought of Sati‘ al-Husri (Princeton, New Jersey: Princeton University Press, 1971).
Adeed Dawisha, Arab Nationalism in the Twentieth Century: From Triumph to Despair, Princeton, NJ: Princeton University Press, 2005
Michael Scott Doran, Pan-Arabism Before Nasser: Egyptian Power Politics and the Palestine Question, Oxford: Oxford University Press, 1999.
Gershoni, Israel and James P. Jankowski, Egypt, Islam and the Arabs: The Search for Egyptian Nationhood, 1900–1930 (New York: Oxford University Press, 1986).
———, Redefining the Egyptian Nation, 1930–1945 (London: Cambridge University Press, 1995).
Hourani, Albert, A History of the Arab Peoples (New York: Warner Books, 1991).
———, Arabic Thought in the Liberal Age. 1798–1939 (London: Oxford University Press, 1970).
Weiner, Andrew Stefan, "Insurgency and Circumspection: The Legacies of Pan-Arabism," Afterall: A Journal of Art, Context and Enquiry, 40(2015): 90-101