The problem with this argument is that going from most of south America to any part of central America would be a large economic jump. And we know that people who come into the USA illegally tend to go towards major hubs with more economic activity and opportunity. The amount of economic migration would increase for sure. And in a situation where we already turn around over 100,000 a month for attempting to cross illegally, not to mention the number we never catch, it's definitely not going to advantage American citizens to have open borders along the Mexico-USA border.
Where there's economic activity and opportunity that immigrants are drawn to, like you said, then that means there's an unmet demand for labor in the host country. A business that could expand, but isn't able to because it can't find workers.
Immigrants then spend the money they've earned which further stimulates the local economy and increases tax revenue for governments.
This basic reasoning is why economists are pretty united in the idea that immigration is good for citizens of host countries experiencing immigration, and real-world data has shown this over and over again.
The idea that immigration to the United States disadvantages citizens just isn't supported by evidence.
>then that means there's an unmet demand for labor in the host country
Where did you come up with that idea? Maybe you meant there is unmet demand for labor at a lower cost, or lower than minimum cost (under the table, etc). This can come at the expense of people earning moderate wages who are not going to be happy about this. A race to the bottom does have issues.
If anything it would disadvantage citizens.