Every dollar invested in the us stock market is a dollar NOT invested in the European one.

This is functionally the same thing.

Europeans sending productive capital to USA instead of their own companies at home.

In practical terms it means that a US business may survive that would otherwise have failed, and an European company might fail that would have otherwise survived.

In economic terms, it doesn't really matter if any actual physical business physically moves.

It matters where resources are being allocated, as this means they are not being allocated elsewhere.