Can the roots of the Iranian Revolution be traced straight back to the 1953 coup?

It's possible to trace the causes of the 1979 Iranian revolution back to the 1953 coup, which ousted Mohammed Mossadegh, but why stop there? That was just a prominent event in a long struggle for dominance between Iranian nationalists, British colonialists, oil companies, and the dominant forces of the Cold War. Focusing on the coup distorts history; you have to look at everything in context.
By the time Ayatollah Khomeini and his followers seized power in Tehran, Iran was a client state of the United States and a proxy for American influence in Southwest Asia. It was perceived as reliable because the Shah was willing to act in favor of American interests in return for money, arms, and a guaranteed market for Iranian oil. Economic and political inequality in Iran, its alignment with America against the Soviet Union, and Iran's tacit support for Israel, even though it voted against the United Nations' recognition in 1947, contributed to the rise of a nationalist movement allied with radical Islamism in the 1960s and 1970s. The 1979 revolution was the culmination of a power struggle between the nationalists and Islamists, the monarchy and the internationalists, and the outside neo-colonial actors on all sides of the Cold War and globalization.
Pointing to the ouster of Mossadegh as the cause of the Iranian revolution misses the point. Recall that the Shah initially opposed the coup because it was wildly unpopular in Iran. He was coerced by the United States and Britain, and he was convinced to support it by his senior army officers as the best way to retain power. After he relented, he ruled repressively, but more or less securely, until the late 1970s.
After the revolution, all the old political fractures remained unhealed. There was still a conflict over oil, now complicated by American sanctions. There was still a nationalist movement, though not allied with radical Islam, competing with the government and enjoying support in the army. There was still Cold War geopolitics, and there was still regional instability, both of which were made worse by the conflict in Palestine and the rising power of Iraq. These external factors contributed to the change in the balance of power inside Iran that made the revolution possible and helped consolidate it afterwards.
https://archive.nytimes.com/www.nytimes.com/library/world/mideast/041600iran-coup-timeline.html?mcubz=3

https://web.stanford.edu/class/e297c/war_peace/middleeast/hiranianrev.html

https://www.mei.edu/publications/iranian-revolution-february-1979