Everything you wrote is correct. However, Mexico is actually an immigration success story. The net migration flow is around zero [1].
The big picture comes down to supply and demand. Today's supply is from specific countries: Venezuela, Guatemala, Honduras, El Salvador, Brazil, a few others. Each country has a different rationale, but generally it boils down to violence, poverty, and Putin. Not necessarily in that order, and often it's all three.
"Demand" is due to the congestion backlog in the US immigration courts. A prospective refugee might not see a judge for a year or two. During this period they have to be paroled in and granted work authorization.
Most applicants today aren't genuine refugees. This was not the case in prior decades because there was no backlog. Awareness of this loophole makes the US a much more practical and appealing destination than it used to be.
The backlog, in turn, stems from the congressional paralysis on immigration. For 20 years the nativists blocked bill after bill, despite large bipartisan support for reform. They did so because every compromise also included a guest-worker program and other immigration benefits.
More recently, there was a deal on the table with no GWP and no immigration benefits. In previous years, it would have been a nativist's dream. It was blocked by the Trump campaign in order to "run on the issue." [2]
A large fraction of the 2024 immigration numbers is due to Trump, maybe as much as 50% or 80%.
For the bigger picture, consider the fact that the exodus in Venezuela and Syria was started by Putin. He gives you the flu (waves of fleeing migrants,) blames the aspirin (the "globalist" Western governments who are forced to handle them,) then sells you the Ivermectin (Trump, Orban, Le Pen, AfD, etc.)
[1] https://www.pewresearch.org/short-reads/2021/07/09/before-co...
[2] https://www.nbcnews.com/politics/congress/senate-republicans...