I prefer option B as the correct answer. The reason is that the adverb "however" helps to indicate that a contrary option is being expressed.

The reason answer choice B is incorrect is that "but" and "however" both indicate a contrast - using both words is redundant.

Can some experts explain what is wrong with options C and D?

"Even though" in D and "despite" in E both indicate that there is a contradiction between being progressive and inefficient - that somehow her progressiveness should be affected by her lack of efficiency. Since efficiency should have no impact on progressiveness, contradiction words are inappropriate here. We instead want a contrast word (like "but" or "however") to indicate that there is a difference between her high progressiveness and low efficiency. We can eliminate D and E.

C is different from A only in word order: "not nearly as efficient" in A vs. "nearly not as efficient" in C. Putting that in our own words, A says that the new mayor is not even close to as efficient as the old mayor, while C says that the new mayor is almost not as efficient as the old mayor. Rephrased like this, we can see that these have two very different meanings, and that C doesn't actually make much sense. We can eliminate C.

This leaves us with A as the correct answer.