Schwab is a fairly common family name in Southern Germany/Austria. It comes from the Duchy of Swabia. There is no one "Schwab bloodline". I've never seen any evidence linking Klaus Schwab to either Marx or the Rothschilds, nor does she present any here.

Schwab is definitely not related to the Rothschilds. That was a meme going around that wont seem to die. It's based on a Mary or Marion Rothschild who married a man name Fred Schwab - I may have the first names wrong - they would be the right age to be Klaus's parents. But they were not. Klaus parents are known. His father was Eugen Schwab.

His mother is a different story, there could be an interesting story there. There is speculation that his biological mother abandoned the family after he was born and it's his father's second wife who Klaus refers to as his mother. The first wife was apparently Jewish and left for America. But her name was not Rothschild, nor Mordecai/Levi/Marx