To understand men of faith, one must understand the origins of faith. Man has two bases for his view and understanding of nature and of himself -- reason and faith. Reason is based in logic and the rational; faith is based in the soul. Logic and faith have different rules and different ways to function, survive, and view the world. Logic understands through hypotheses, from which it derives rules. Logic cannot tolerate contradictions and draws distinctions between approximations and absolutes. Logic is common to all humankind; what an African person understands through logic will also be understood by a Chinese person through logic and vice versa.
Logic and reason have methods and order common to and accepted by all, just like all share hearing and sight -- the rainbow can be seen by Mongol, African, Jew, and Muslim. So, too, will all deduce the same laws of physics.
In contradiction to logic, by its nature the soul has its own rules. The soul can tolerate contradictions, loves vagueness, longs for the secret, and it is highly doubtful anyone could formulate a consistent and orderly method which works for the entire human race. The well-known saying "There's no accounting for taste" is doubly true when it comes to what amazes the soul: "There's no accounting for envy, desire, lust for honor, or faith." Faith, which is based in the soul, belongs to the individual, to the private domain -- what the Chinese believes the African will reject, and vice versa. That is why academic institutions, based on logic and reason, can develop in tandem throughout the world and pass information from one to the next, understand and accept common logical information. Synagogues, mosques, and churches not only cannot work together, they despise each other; the absolute truth believed in by a Muslim is apostasy for a Christian, the Christian's absolute truth is the Jew's heresy and vice versa. Each person has his own unique soul.
And now to your question: Chazal did indeed take pride in their magic abilities: "Anywhere the sages looked -- they brought either death or poverty" (Moed Katan 17b). R' Yochanan got angry at a student who questioned the word of the sages and turned him into a mound of bones: "You are sneering at the words of the sages!' He set his eyes on him and the student turned into a mound of bones" (Bava Batra 75a). On the other hand Rabbi Yehuda son of Bava, one of those martyred by Rome, stood still as a stone in the face of the Roman soldiers and said "I am before you as a stone which cannot be overturned…They drove three hundred iron spears into his body and made his corpse like a sieve!" (Avodah Zarah 8b). You should not be puzzled at the contradiction, as you yourself sensed, for faith can tolerate and even longs for contradictions. As faith is more questionable, it elevates the soul of the believer to new spiritual heights (that is, imaginary hallucinations). As the believer obeys ever more absurd demands, so his faith is thought to be deeper. In the jargon of the believers: "He stood the test." That is why G-d and the rabbis can play with the soul of the believer, and the more they play the more entrenched in the believer's soul will be greatest of absolute obedience and blind faith.
Go learn from our forefather Abraham, who was tested ten times and stood through them all. The last two of these tests were the expulsion of Hagar and his son, Ishmael, and the bringing of his son Isaac as a sacrifice.
What was the subject of these tests? His ability to obey! In contradiction to his reason, understanding, and the stirring of his conscience. For his first test he was ordered to banish Hagar and his own son Ishmael.
"And Sarah saw the son of Hagar the Egyptian, whom she had borne to Abraham, scoffing. Therefore she said to Abraham, 'Cast out this bondwoman and her son; for the son of this bondwoman shall not be heir with my son, with Isaac.' And the matter was very displeasing in Abraham's sight because of his son. But G-d said to Abraham, 'Do not let it be displeasing in your sight because of the lad or because of your bondwoman. Whatever Sarah has said to you, listen to her voice; for in Isaac your seed shall be called. Yet I will also make a nation of the son of the bondwoman, because he is your seed.' So Abraham rose early in the morning, and took bread and a skin of water; and putting it on her shoulder, he gave it and the boy to Hagar, and sent her away. Then she departed and wandered in the Wilderness of Beersheba."
The Scriptures specify that Abraham objected to the banishment of his son, but that G-d requires the banishment, arguing "for in Isaac your seed shall be called." What does Abraham do? Obeys and hurries to fulfill the Divine command: "So Abraham rose early in the morning."
Immediately after Abraham is asked to act contrary to what might be expected after this test. Just after Abraham banishes Hagar (his mate) and his son, using the excuse that Isaac will be the one from whom his lineage will descend, G-d commands him to kill his son Isaac, the son for whose sake he banished Ishmael. What does Abraham do? "So Abraham rose early in the morning… And Abraham stretched out his hand and took the knife to slay his son." Absolute obedience!! Against his logic, against his conscience, and against the promise for his future. This is what faith demands: obedience to the unimaginable, to the irrational, to the unexpected. As the totality and absurdity are greater, so is the faith considered loftier.
In light of the above, the answer to your question is clear. Not only does the believer not fear contradictions, contradictions raise his own self-perceived level of spirituality. In other words: The power of faith finds expression in things which contradict reason and deny the reality which is apparent to him.
Sometimes people of faith do use their reason, but this use is as support and aid to faith. As the philosopher, Benedict Spinoza, wrote: Reason is a handmaiden to faith, meaning that logic is a helpmeet to faith. Instead of reason challenging faith and contradicting it, men of faith strengthen their faith to emphasize its power against reason and to demonstrate their faith.
To understand how they partially use their wisdom as handmaiden to their faith, read the essay Dealing with the Contradictions Between Torah and Reason.