Could 6 tubes of blood draw over 3 weeks quickly worsen anemia symptoms of a severely anemic person (half the blood volume, half the rbc)?

No. : The amount you describe is negligible. Total blood volume is about 5600 ml in an average adult male, and 6 tubes if blood would be about 60 ml, just over 1%. The amount is too small to have an impact.

No: An adult has roughly a gallon of blood flowing through their veins (and arteries and, to a small extent, the lymphatic system). They would have to be very large tubes. I would have to assume that they are drawing blood because someone is trying to identify a cause of illness and that illness is also making the person anemic.