NAFTA: North American Free Trade Agreement.

TN-1: a status that (in theory; I defer to your experience) that allows Canadians to show up at the border to be approved to work in the USA.

Ergo free trade of labor and products are the same.

So why are citizens of the signatories theoretically allowed free movement among the three countries of NAFTA? What is the purpose of the free movement of labor in NAFTA? Why is the free movement of labor integral to free trade?

The free movement of labor facilitates free trade. The purpose of NAFTA is free trade of stuff, not labor. The signatories understood that certain classes of labor facilitated and contributed to free trade. They knew that without this free movement of certain skilled classes the provisions for trade among the signatories would be hampered.

Countries can not trade labor. The government can not trade what it does not own. Free trade among nations is not nations moving stuff back and forth freely. It is releasing the binds of government that prevents free trade from occurring among individuals and businesses in the countries that are part of the agreement.

Yet, all of the countries involved in NAFTA have determined that it is in their interest to know who is entering the country, why they are in the country, how long they will be in the country and how much money they will make while in the country. They can and will ask those questions in determination of whether the person has a legitimate NAFTA exemption to move freely from country to country.

The receiving country makes the determination. A citizen of one of the signatory nations can not unilaterally determine that "Oh, NAFTA! Let's go to Canada." They must still seek permission and acceptance of their exclusionary status under NAFTA.

It happens when you cross the border at a point of entry. I have done it many, many times.

"Why are you coming to Canada."

"I am coming under NAFTA to work based on......"

"Where will you be working?"

"....."

"How long will you be in Canada?"

"....."

"How much will you be making for your work?"

"......"

Do you have a work permit?"

"No, but I will get one if needed."

"Take this paper to the immigration officer and they will finish your entry requirements."

You see, I have to ask Canada's permission to enter their country based on a defined set of circumstances. They decide, not me. Even under NAFTA. Especially under NAFTA.

Your own words:

If the approval is not granted, they stay in Canada.

If you are interested, an overview of the NAFTA exemption is here (warning: obtuse governmentese language)

The detail from the actual treaty is here. Warning, even worse, it is the actual treaty.

Chapter 16, Section 1603 with annexes is the relevant chapter.