Did the idea of infants with original sin emerge from the early West and from St. Augustine?
No. It came from divine revelation and is found clearly in Sacred Scripture. There is,
agreeably, a different theological approach to the matter in East and West, but the doctrine is actually the same.
St. Augustine reflected on original, or ancestral, sin extensively and profoundly, and his treatment of this teaching has been by far the most impactful in the theology of the Church.
This being said, some of the Eastern Fathers treat of it as well, such as St. John Damascene, St. Maximos the Confessor, St. Gregory of Nyssa, and others. It is crucial not to pay too much attention to the dichotomizing approach of some Eastern thinkers today who make far too much of the difference of approaches. Eastern theology tends to stress personal ascetical struggle, and Western the universal fallen state of man. These are complementary, not contrarious, perspectives.