Your model of the southwest border may need to be updated because the dynamics have changed a lot in the last 4 years. Nowadays people cross the border and intentionally get caught by border patrol (in many cases by calling 911) so that they can make asylum claims. Even if they are arrested, they are generally released and allowed to live in the US for many years while their cases are pending. The number of people doing this has shot up during the last 4 years and is now about 3 million people per year [1]. By comparison, the number of visa overstayers is less than 1 million per year. [2].

[1] https://www.cbp.gov/newsroom/stats/nationwide-encounters [2] https://www.dhs.gov/sites/default/files/2023-07/23_0707_FY22...