What health-care system was founded by andrew taylor still, an allopathic physician who disliked the toxic drugs of his time?

That would be the practice of Osteopathy. This practice was started by Andrew Taylor Still who also coined the term "osteopathy". The practitioners of osteopathy emphasize in diagnosis and prevention as well in treatment the interrelationship between various body systems as well as the physical manipulation and manual adjustments of bones and tissue.

Andrew Taylor Still is considered the founder of osteopathic medicine. He was also a physician, surgeon and author. He pioneered the idea of "wellness" and would be considered a forerunner of the concept of wholistic medicine. He followed in his fathers footsteps and became a doctor in Missouri back in 1860 and was a surgeon during the civil war. He became very skeptical of traditional medical practices when three of his children died from spinal meningitis. He spent many years studying the human body and dedicated himself to finding better ways to treat illness and disease.