It should be pretty clear why Mexico wants more illegal immigration to America:
Diplomats from Mexico and Central America on Monday demanded guest worker programs and the legalization of undocumented migrants in the United States, while criticizing a U.S. proposal for tougher border enforcement. ...Mexicans working in the United States are a huge source of revenue for Mexico, sending home more than $16 billion in remittances in 2004, Mexico's second largest source of foreign currency after oil exports according to the country's central bank.
Mexico's GDP is approximately US$1 trillion, so 1.6% of their GDP comes from immigrants in America who send money home. What's more, there's no way to account for the indirect benefits that Mexico receives because these immigrants are supported by the American health care system, education system, and infrastructure. Finally, many illegal immigrants (not all, or even most) were criminals and thugs in Mexico, and I'm sure the Mexican government would prefer them to prey on Americans rather than Mexicans and to fill American jails rather than Mexican jails.
It's interesting to consider, however, whether Mexico would really want a completely open border. Without all the restrictions, which Mexican citizens would be most likely to come to America? Perhaps the best educated and wealthiest, rather than the poorest?
In response to your last paragraph, the wealthiest and best-educated Mexicans probably would get a visa if they applied for one. The Mexicans crossing over ilegally are those whose visa applications would be denied if they applied for one (which some of them might have). The ability of upper-class Mexicans to come into the US is not what Mexico is concerned about. I don't think Mexico has anything to lose from having an open border.
What Mexico apparently wants to do is somehow convince the US that people who broke US law while crossing the border ought to have this crime simply ignored by US authorities. This would not only pardon illegal immigrants and thus retroactively turn them into legal ones, but also lower the standards regarding who is allowed into the US. American consulates abroad do have standards about who gets visas and who doesn't - and these standards are hardly "the tired, the poor, the homeless, the tempest-tossed, the huddled masses yearning to breathe free, the wretched refuse of your teeming shore", or however it goes.
To me the most absurd part of all this is how the Mexican president has said that, if the US protects the border more actively or builds a wall, then this supposedly disrespects migrants' rights. I guess there must be something in the Mexican constitution that says all Mexicans have an inalienable right to break US immigration laws...
And this is coming from am immigrant. An immigrant who learned the language and got a visa. It's not that hard. Sure, I understand that the Mexicans who migrate illegally into the US probably do not have the education or financial resources I have. However, "my country's economy sucks" hardly justifies breaking the law, or being a refugee. And even it if does justify breaking the law on basic moral and ethical grounds (yes, I am a soft liberal at heart, who believes criminals are often - but not always - led to crime by their environment), I still don't think criminals should get away with breaking the law. (I think speed limits are too slow and I break them, but if I am caught, I pay the price).
(And yes, I as an immigrant am aware of how my views on immigration may be reminiscent of the Russell Peters bit about how his father, an Indian immigrant, starts to turn into a redneck after many years in the US, saying bad things about immigrants who don't learn English...)