Eliyahu was answered as soon as he prayed. How does that prove the importance of mincha over other prayers?

The gemoro in Berachot 6b says,

Rabbi Chelbo said in the name of Rav Huna: one should always be careful about the prayer of Mincha, as Eliyahu was answered only at the time of the prayer of Mincha, as the pasuk says: “And it was at the time of the afternoon offering that Eliyahu approached and said, ‘answer me, Hashem, answer me.’”

Translation from here.

The gemoro almost gives the mistaken impression that Eliyahu started praying before mincha and was only answered at mincha.

But the narrative in Kings 1 18: 25 etc. seems to show that it was only at mincha that Eliyahu started his “operation” and uttered his prayer. That is, he was answered as soon as he prayed. See the text below slightly adapted from here.

25 And Elijah said to the prophets of the Baal, "Choose for yourselves the one bull and prepare it first since you are the majority, and call in the name of your deity, and fire place not."

26 They took the bull that he gave them and prepared [it]. And they called in the name of the Baal from the morning until noon, saying, "O Baal, answer us!" But there was no voice and no answer, and they hopped on the altar that they had made.

29 And as noon passed and they feined to prophesy until the time of the sacrifice of the [afternoon] offering, and there was no voice and no answer and no one was listening.

After that Eliyahu set up his altar and prayed.

If he would have set up the altar at another time, he might well have been answered then too!

Do we have to conclude that he was after all answered at mincha and therefore mincha is important despite the fact that he did not try at any other time?

See related What is special about praying Shacharis and Mariv (as opposed to Mincha)?

Answer

Check out the Rif (not the halachic commentary) on the Ein Yaakov. He asks this question and brings several answers.

One of the answers he gives is that Eliyahu may not have deserved to be answered at all, since he was offering a sacrifice outside of the Beit Hamikdash (even though it was permitted to him, it still was slightly connected to a sin - See Rashi Bereshit 35:11 where Eliyahu's offering of the sacrifice is referred to as "actions of the nations"). However, since he prayed at Mincha, the time when Yitzchak was being offered as a sacrifice, the merit of Mincha helped him out and his prayers were answered.

Hence we see the importance of the Mincha Prayer from the fact that Eliyahu was answered at Mincha.