He was not appointed chairman of the committee, being considered not very popular with all its members.是因为表示原因，所以采用being considered吗？

——理解正确！

还有这句话：Being taken good care of，he recovered quickly。（表原因？）是不是也可以改为Taken good care of，he recovered quickly。

——理解正确。

前者暗示， recover 的动作逐渐发生在“被照料的过程中”；后者指“被照料发生在先，且已完成，明显的原因状语”。Taken good care of，he recovered quickly. = Having been taken good care of，he recovered quickly。