"Apart from the idea of sorting people into useful and useless being inhumane, it also seems to be counterproductive."
I'm sorry but this is totally unfair.
Nearly every nation has immigration criteria.
Canada, Australia and UK (non-EU) all have a 'points system' which definitely separates 'the useful' form the 'useless'.
Education, skill-set are key components.
I would argue language ability should be added to the list as it's the #1 predictor of integration. It's impossible to engage with a community that one cannot communicate with - and this will help a lot with those immigrants who are likely to be most marginalized (i.e. not the rich one's we are worried about). Language training should be a primary part of social support for newcomers.
"It looks like every kind of screening of immigrants will deter the more desirable ones"
No, this is not true at all. The most talented immigrants are entirely undeterred. The nations noted above are good examples of that.
"as far as that determination is possible on their arrival at all."
There is no 'determination' at arrival. The discriminating criteria are usually applied before immigrants arrive.
This doesn't preclude status for refugees and other family-class migrants either - you can have both.
Finally - I think looking at success rates of immigrants overall is a much better exercise than looking at a handful of billionaires, though the later does have value.