The answer is it depends on religion.
To answer the questions very fast...
Q: Is religious fanaticism a necessity?
A: No, actually with current values it would only make things worse.
Q: Is a strict and conservative view the only way to prevent corruption?
A: Most likely yes. But not through Faith, but through Cardinal Virtues, Respecting the Ten Commandments, Seven Noahide Laws and other similar spiritual views.
Q: How much power would the Church wield in the daily lives of people?
A: If it goes 'all in' in Faith, and no methods to apply it's Cardinal Virtues, then it would go in pure dictatorship, poverty and slavery. If it applies Cardinal Virtues, most likely a lot of power at the beginning, then it's monetary power will fade away, probably a lot more than in the present time. The credibility would probably stay intact or increase in that case.
First, Catholicism, and as a matter of fact, almost all Christian religions never really used or promoted the Four Cardinal Virtues. Probably because, if they did, there would be less Faith in the Church, and the Church would lose power.
Instead, by overly promoting theological virtues like Faith, Hope and Charity, without cardinality, the Church mostly accomplished the ability to increase it's revenues, instead of actually helping people.
Theological virtues only work, if they are very well backed by Cardinal Virtues.
If theological virtues are not backed by cardinal virtues, it is very similar to how a currency is not backed by gold.
And in regards to the question, if say, any religion would promote, and succeed promoting Prudence, Fortitude, Temperance and Justice, no people would be harmed by some hypothetical 'demons', and as a matter of fact, problems like world hunger, people being poor would be solved in single digit years. And who knows, after that, serious space exploration?
But that would also mean, the end of bread and circuses.
Anyone who would try doing space exploration while keeping bread and circuses would attempt to collapse the Universe, but that's a subject for another story.
The only other religion known that would pursue a form of Cardinal Virtues, is Buddhism. With it's Noble Eightfold path. Of course, there is Tenfold Path too.
Whether you have the Noble Eightfold path or the Cardinal virtues, they are sending essentially the same message. Avoid the Seven Deadly Sins.
Of course, Buddhism was first, 2500 years ago. Cardinal virtues came later through the works of Plato, in it's work, The Republic.
Prudence or Wisdom is called to be the Mother of all Virtues.
So, to answer the question, there is no need for any religious fanaticism that would imply values like 'Faith, Hope and Charity'. Because if it did, it would only keep the Church's coffers more full, thus indulging in greed, and making more ficticious 'demons' happy.
If somehow, 'demons' would appear and drive the world crazy, and Faith would be promoted instead of Prudence, and the Church is here to save you with their faith, then, most likely, the Church is the culprit behind it.
If, however, a religious fanaticism, that would imply the application of Cardinal virtues or the Tenfold path(Because Ten sounds better than eight), or even better, the application of the Ten Commandments or The Seven Laws of Noah(Noahide Laws), then it could not be called 'fanaticism' anymore, but become the new normal.
To call a 'fanatic' someone who would want to be Just, in God's eyes, is blasphemy.
"Love justice, ye that judge the earth"
Love works with Cardinality. Dante Aligheri with the terms Deficient Love, Excessive Love, and Malicious Love, define the Seven deadly sins.
Allegorically, the Purgatorio represents the penitent Christian life.
In describing the climb Dante discusses the nature of sin, examples of
vice and virtue, as well as moral issues in politics and in the
Church. The poem outlines a theory that all sins arise from love –
either perverted love directed towards others' harm, or deficient
love, or the disordered or excessive love of good things.
The core seven sins within Purgatory correspond to a moral scheme of
love perverted, subdivided into three groups corresponding to
excessive love (Lust, Gluttony, Greed), deficient love (Sloth), and
malicious love (Wrath, Envy, Pride).
Faith, Hope and Charity attempt to guide humanity to love. But with no education to not practice deficient, excessive or malicious love, it is all for nothing. This is why Prudence and Justice are the cure for Malicious love, Temperance is the cure for Excessive love, and Fortitude is the cure for Deficient love.
The planet Venus (the Morning and Evening Star) is traditionally
associated with the Goddess of Love, and so Dante makes this the
planet of the lovers, who were deficient in the virtue of temperance
Regarding the Church: St. Peter then denounces Pope Boniface VIII in
very strong terms, and says that, in his eyes, the Papal See stands
empty (Canto XXVII).
Within these circles Dante can discern the human form of Christ. The
Divine Comedy ends with Dante trying to understand how the circles fit
together, and how the humanity of Christ relates to the divinity of
the Son but, as Dante puts it, "that was not a flight for my wings".
In a flash of understanding, which he cannot express, Dante does
finally see this, and his soul becomes aligned with God's love:
But already my desire and my will were being turned like a wheel, all
at one speed, by the Love which moves the sun and the other stars.
Through Cardinal Virtues(the equivalent of a compass), Love, Faith, Charity, Hope can be embraced.
Edit: Wiki links are now clickable.