Why the word being is included in this sentence “she eliminated her without a second thought as not being to her husband's taste”?
There are two women involved, I'll write her¹ to mean the subject of the main sentence and her² to mean the other woman, the one who is eliminating.
“Not being to her¹ husband's taste” is a description of “her²”. The complement “… as not being …” gives the reason for eliminating her².
The sentence could also be written “She¹ eliminated her² without a second thought because she¹ decided that she² was not to her¹ husband's taste”, except that there is a collision of referents for “she”, so an alternative description would have to be used for at least one of the women (e.g. “She eliminated Alice without a second thought because she¹ decided that Alice was not to her¹ husband's taste”). The construction with a present participle is more direct and is not ambiguous.