Definitely Not A Cop said:
Am I understanding what happened correctly?
India wins independence. India decides it will partition off a portion of itself for Muslims to live freely*, to be called Pakistan. Muslim residents still living in India begin facing intensified discrimination and violence. The train scene in the show is representative of the millions of Muslims trying to escape India to Pakistan.
*Assuming the intent wasn't really to allow them to live freely, but to get rid of them.
From someone who thinks "both sides" is usually a cop-out... this was very much a "both sides + also the British" problem.
India's independence movement began to fracture in the early 1900s on several ideological lines. The most important preface to the whole Indian subcontinent is that "Indian" and "Pakistani" nationalities are myths. They are modern creations. In reality, you have a patchwork of many, many cultures, which you can very broadly divide into North Indian/Indo-Aryan (Pakistan, Bangladesh, and the upper half of India) and South Indian/Dravidian. All these cultures have different languages, different cuisines, different folk tales and heroes.
Muslim leaders, especially Jinnah, became convinced that without any real national unifying force, Indians would rally around religious lines - people would start identifying less and less with their ethnic group and more and more around religious identities. Centrist Indian independence leaders like Gandhi and Nehru insisted that a secular India and a secular Indian identity was very much possible, but there were growing movements among Hindu Indians that were pushing a national identity based on adherence to the Dharmic faiths (Hinduism, Buddhism Jainism, sometimes Sikhism) and decried both major Indian Abrahamic faiths (mainly Islam but also Christianity) as foreign and unwanted - despite the fact that the Indian versions of Islam and Christianity are pretty unique and well-blended into Indian cultures, which is why Indians and Pakistanis living in the West tend to get along pretty well generally.
Eventually, Muhammad Ali Jinnah and the other Muslim leaders pushed for the creation of Pakistan, or at least a kind of federation system in India that would protect Muslim rights. In 1946, a plan was put forward that would divide India into many different states. These states would form 3 super-states with strong self-governance - these super states were basically the modern India, Pakistan, and Bangladesh. The three super-states would together form a national union which would oversee free trade, currency, foreign affairs, and the army.
Jinnah and the Muslim leaders agreed to this plan because it provided satisfactory protections to the Muslim minority, and so did the Hindu leaders because it kept India together. However, the British were in a hurry to form an interim government and handed off power to Nehru and the centrist Indian National Congress. Nehru, although he wasn't biased against Muslims, was a leftist who wanted a strong federal state so he could quickly industrialize the country. The British bypassed the main Muslim party despite assuring Jinnah that they would have a seat in the government. Then, Nehru made a speech disavowing the three-part structure plan that was drawn up, stating that the states would not be grouped into the super-states (the quasi India/Pakistan/Bangladesh). Jinnah and the Muslim League saw this as a betrayal and a sign of things to come for India's Muslims and completely abandoned any attempt at unity, insisting that an independent nation for India's Muslims was necessary for survival.
As a result, Partition. The British bungled that as well, letting the former vassal Indian rulers decide which countries they wanted to join despite the fact that said rulers were not always the same religion as their subjects (this is the root of the Kashmir conflict). Millions of Muslims suddenly stuck in India had to leave their homes, but so did millions of Hindus and Sikhs who were now stuck in Pakistan. The show, because it focuses on a Muhajir Muslim family, showcases the Muslim side of the tragedy, but Hindus and Sikhs in Punjab and Sindh also suffered greatly.
I think the 1946 Plan might have been the subcontinent's best chance at unity. I don't really know if Jinnah was right that all of India's Muslims are one nation - the separation of Bangladesh and the struggles of Pakistan sort of attest to that - but I think he hit the nail on the head that eventually, India's identity would begin to take on more and more of a Hindu identity which could be to the detriment of Muslims.