In spite of all their shortcomings, we cannot ... admire them.
Answer Details
The correct word to fill in the blank is "but".
The sentence expresses that although the subject has some flaws or negative qualities, we still find them worthy of admiration or respect. The word "but" is used to indicate a contrast or opposition between the two parts of the sentence. It acknowledges the shortcomings of the subject, but goes on to suggest that there are other qualities that outweigh those negative aspects and deserve admiration.
For example, we might say "She's not perfect, but we can't help but admire her hard work and dedication." Here, we recognize that the person has flaws, but we still find them admirable because of their other positive qualities.