You answered your own question. LOTS OF PEOPLE WANT TO COME.
Where does it end? We have legal immigration and that's good enough. Numbers larger than what we let in now make it impossible for the nowcomers to assimilate.
Heaven forbid we repeat Elise Isle. Look at all the people we took in that eventually assimilated! It took a few generations to get there.
At this point it is not the immigrants per se that are the problem, it's the power hungry politicians who have worked extra hard to convince these people that they DON'T have to assimilate (Diversity, the lefts form of Godliness) and in doing so you will get extra perks. Useful idiot stuff.
It just seems to me that with some strict real rules and common sense this should be workable