From what I can tell, a big part of the problem in Europe is that people seeking asylum are prohibited from making a living (due to widespread belief in the lump of labor fallacy) and so have to be dependent on welfare.

Yes and when a government tries to do something about that (like Spain granting temporary permits so they can work and pay taxes) it angers the right even more.

>asylum are prohibited from making a living

This is exactly how we got here. We allowed them to work to make temporary asylum permanent migration.

Look, you can either let people make a living and manage their own lives while contributing to society, or you can not let them make a living and ensure they're housed and fed. You can't be angry at A and also be angry at B.