I don't know how hard it would have been to get to China for an American citizen in the 1960s, but I know that American citizens weren't allowed in the Tibetan Autonomous Region at all until the first tourist visas were issued in the 1980s.

The primary reason I'm sure was the fact that the People's Republic was not internationally recognized, most importantly by the United States which recognized the Republic of China (i.e. Taiwan) until Nixon's visit to China in 1972. As it concerns Tibet, the United States funded and armed the Four Rivers Six Ranges (Tib: chu bzhi gangs drug) group which started as a Revolutionary Khampa organization, and evolved generally into a Tibetan freedom group as the war in Kham evolved into a more general struggle for Tibetan autonomy, not just in Kham (east Tibet). The United States' support for 4R6R dried up almost immediately upon Nixon going to China. Makes sense. The whole point was to form an alliance with China against the USSR, and this meant not arming a domestic enemy of the Beijing government.

Ironically, if an American did have access to China of the 1960s, they most likely would have been able to travel to Kham, where 4R6R was primarily based, since it half of it was absorbed into Sichuan Province (where regular Chinese visas tend to apply). Though my guess is that American visitors to China at this time were rare and difficult, though I can't say to what degree. And American would not have been able to legally travel to Tibet at this time without special exception from the Beijing government. I am not sure of any incident of this happening.

Sources:

Warren Smith, Tibetan Nation: A History of Tibetan Nationalism and Sino-Tibetan Relations

ibid., China's Tibet: Autonomy or Assimilation