By that standard why not open our borders to the world? Let everyone come in. Why be so partial to the Mexicans? There are billions of people around the globe that yearn for freedom. Does being nation of immigrants mean that America should not control immigration even if most Americans want such controls?
Having said that let's take a moment to recall the history of immigration laws in America. The fact is, they were purely racist inventions (demanded by the people as you would put it), intended to keep the "Yellow-Peril" from taking our jobs, our women, our country. (The same kind of rhetoric we now hear regarding Mexicans.)
Chinese first immigrated to America in large quantities when reports of the California Gold Rush reached coastal China in 1849. Immigration reached its gold rush peak in 1852, when over 20,000 Chinese, mostly farmers from around the Canton area, headed over to work mines in search of gold. The immigration slowed drastically afterwards, until the late 1860s, when Chinese papers advertised looking for workers on the railroad, and the rush was on again.Those Chinese who still worked various private mines became the outlet of white anger, and blamed for lost jobs. In 1854 the California Supreme Court ruled that the Chinese could not testify in court in any case in which a white person is a party. The threat of the Chinese to the working class and their jobs continued to be a constant theme up through the early 20th Century.
Having been driven out of mining and agriculture, and laid off as work on the trans-continental railroad came to a close, the Chinese immigrants moved into other work, such as manufacturing, laundering, and domestic occupations, running head first into another minority group: the Irish. The Chinese would often take lower wages than the Irish workers, and many employers found them by and large to be a far superior working group to the Irish, cleaner, more hard working.
But the leaders of the Irish community took the opportunity to attempt to raise their own status in Anglo-Saxon society, by promoting a sort of pan-ethnic whiteness, defining Irish and Anglo-Saxon peoples to stand together in a 'white' category, as separate from 'black' or 'yellow' races. They used the imagery of the Yellow Peril -- legions of Chinese sweeping into the country, taking away the good honest work of the white man. They were for the most part very successful.
American legislators became obsessed with stemming the oriental tides that they feared would soon overtake them. In 1790, the Naturalization Act explicitly stated Naturalization as a citizen was only possible for "free white persons" only. This did not necessarily exclude Asians, as many people considered the Asiatic races to fall into the 'white' category (at least, George Washington did). In 1870, the abolition of slavery prompted a change in the wording, and it was amended to include persons of African descent. It was also amended to specifically exclude persons from China.
In 1882 the Chinese Exclusion Act was passed, banning not only Naturalization of peoples from China, but immigration as well. It allowed for some loopholes, which were quickly closed up with an 1884 amendment. Ironically, just a couple years later, the Statue of Liberty is unveiled in New York City.