No. If western countries let in 'everyone' who wants to come, they'd be horribly, horribly over-populated and huge swaths of the world would never develop.

The problem is (if you like game theory), that it's in everyone's personal interest to migrate somewhere richer. However, if everyone does that, wherever they came from experiences such massive drain that it'll be worse off. From a purely pragmatic standpoint, unlimited immigration is a poor 'equilibrium' for all involved. 'Developing' nations will never develop, and developed nations will face massive over-crowding.

The reason the west currently allows immigration in the numbers we do is because it's an easy way to get skilled, motivated labour - at once allowing for economic growth as well as driving down the relative cost of labour.

Freedom to travel/migrate should be a human right, however there are many problems with that, considering the current state of the world.

Which is why all of America lives in Atherton and New Haven and nobody is left in Mississippi or Alabama.

I know you're being sarcastic, but California and NY combined have nearly 20% of the population of the US.

In the US, the most prosperous states are definitely also the most populous.

I think that is the justification for allowing in refugees and asylum seekers (i.e. protecting them from human rights abuses), but not usually used for economic migrants.

If that was the justification, then a lottery doesn't achieve that. Nothing short of completely open borders achieves that.