>> Illegal immigration is a market response to a failure of the government to provide a legal framework for sufficient legal migration.
No, it really is not. There's really no upper limit on how many people might WANT to immigrate to your country (other than the obvious 8B world population size), but there's an upper limit on how many immigrants your country can accommodate in the given time frame. Increasing limits for legal immigration won't stop illegal one, it might even encourage it (e.g. when one person gets an immigrant visa and some of their relatives don't but they still decide do go, illegally)