The history of Tibet in the 20th century is very politicized, so I believe that it's best to focus on the primary sources for a more objective understanding of the history of Tibetan independence and China's motivations for controlling Tibet.

The Qing dynasty exerted influence over Tibet through ambans representing the Emperor in Beijing. The power of the ambans varied throughout the duration of the Qing, but had declined significantly by the end of the 19th century. In 1909, the Qing sought to reassert its control over Tibet by sending in Chinese troops. The 13th Dalai Lama fled to exile in India as the advancing Qing army entered Tibet. The Qing wanted to depose the 13th Dalai Lama, but realized that searching for a replacement would be difficult. Instead, they offered to return the Dalai Lama to Tibet as a subordinate figurehead with no political power.

But all of this became irrelevant with the Xinhai Revolution and the collapse of the Qing dynasty in 1911. The newly declared Republic of China (ROC), having overthrown their Manchu overlords, worked to re-envision China from an empire to a modern nation-state. One of core principles of the new Republic was to conceptualize the former empire as a multiethnic Chinese nation composed of "five races": Han, Manchu, Mongol, Muslim, and Tibetan. In Chinese, this multiethnic nation is known as zhonghua minzhu. An early flag of the ROC represents each of these "races" with different colored stripes (the black stripe symbolizes the Tibetans). This idea of a united and harmonious multiethnic Chinese nation would be the primary motivating factor for future interventions in Tibet. As the last Emperor of China abdicated, he declared:

Let Yuan Shikai organize with full powers a provisional republican government, and confer with the Republican army as to methods of union, thus assuring peace to the people and tranquility to the Empire, and forming one Great Republic of China by the union as heretofore of the five peoples, namely, Manchus, Chinese, Mongols, Mohammedans, and Tibetans, together with their territory in its integrity.

Yuan Shikai, the Provisional President of the ROC, wanted to integrate Tibet into the Republic as one of five races "united in one family". He sent the following letter to the 13th Dalai Lama, who was still in exile:

Now that the Republic has been firmly established and the Five Races deeply united into one family, the Dalai Lama is naturally moved with a feeling of deep attachment to the mother country. Under the circumstances, his former errors should be overlooked, and his Title of Loyal and Submissive Vice-Regent, Great, Good, and Self-Existent Buddha is hereby restored to him, in the hope that he may prove a support to the Yellow Church and a help to the Republic.

The Dalai Lama responded by saying that he had no interest in returning to his role solely as a religious leader, and intended to obtain both political and spiritual authority over Tibet. When the Dalai Lama returned to Tibet, he issued a proclamation that essentially, but not formally, declared that Tibet was now independent from China.

On my arrival in India, I dispatched several telegrams to the Emperor; but his reply to my demands was delayed by corrupt officials at Peking. Meanwhile, the Manchu empire collapsed. The Tibetans were encouraged to expel the Chinese from central Tibet. I, too, returned safely to my rightful and sacred country, and I am now in the course of driving out the remnants of Chinese troops from DoKham in Eastern Tibet. Now, the Chinese intention of colonizing Tibet under the patron-priest relationship has faded like a rainbow in the sky.

Qing soldiers were expelled from Tibet. By 1913, the breakaway regions of Mongolia and Tibet mutually recognized their independence from China:

Mongolia and Tibet, having freed themselves from the dynasty of the Manchus and separated from China, have formed their own independent States, and, having in view that both States from time immemorial have professed one and the same religion, with a view to strengthening their historic and mutual friendship.

The ROC leadership did not recognize the independence of Tibet. The Republic was founded as a union of different races under a single nation. Letting Tibet go was tantamount to the disintegration of the Chinese nation. When the People's Republic of China (PRC) replaced the ROC in Mainland China (the ROC retreated to Taiwan), the PRC inherited the ROC's notion of a united Chinese nation, but recast the narrative into a Marxist framework.

With the exception of Mongolia, the international community did not recognize Tibet's independence. Britain was reluctant to recognize an independent Tibet, anticipating that it would destabilize their relationships with Russia and China. The British feared that the Tibetans would ally with the Russians. But for the next four decades, Tibet was de facto independent from China.

The excerpts of primary sources come from Melvyn Goldstein's A History of Modern Tibet, 1913-1951, Sam Van Shaik's Tibet: A History, and other secondary sources.

Another related question: how independent was Tibet on the eve of the invasion, and for how long had that amount of independence persisted?

As a follow on, I had a professor once state that a major part of the invasion included securing water heads for major rivers in both China and India, but he didn't go into detail about it. I'd love to know more about how that might have influenced it.