This is a story that does not apply to our times. There've been many regular scientific revolutions even in the last few decades, the Church hasn't persecuted scientists, and the modern scientific consensus has followed with the revolutions. Whether it's the matter of the cause of the Cretaceous–Paleogene extinction, the revolutions in molecular biology, revolutions in astronomy, or other areas where advances are regular, those outliers actually advancing science have only briefly been ahead of consensus in the real world, but most outliers are cranks, and only those with an expert training in a discipline are likely to be able to identify real advances over crank science. The Galileo model of believing outliers virtually always leaves you in the wrong unless you are a domain-specific expert on the topic.
Look at using MRNA as a medical treatment as another example of a recent revolution, one that's now available in a safe, effective vaccine form.
Scientific consensus provides a means for bad science/scientists to escape ostracism (I use this with the full intensity of its definitions). Because it provides a thick and opaque veil behind which the clergy (all scientists) instructs the laity (the rest of us). Individual scientists no longer stand and fall on their own. So call them cranks if you want but the individuals standing on the banks of the "scientific consensus" stream are arguably the closest we have to 16th and 17th century scientists tinkering away in their private labs, making discoveries that continue to bear their names.
Now, to a philosophy of science I subscribe to: Popper's Falsifiability Test. An undercurrent (probably prominent, even) theme is this: new confirmatory experiments add nothing new if they don't deviate materially from the original experiment. Following this, the whole enterprise of scientific consensus stands on shaky grounds since it adds nothing. It transforms science into a faith-based exercise (akin to bishops at a synod throwing their weights behind certain theories and declaring other bishops as cranks/heretics). This I do not accept.
To my knowledge theories proposed that way in modern science aren't taken seriously and in our present time are not the norm for any recent major scientific discoveries or research. Those days are long past. New science is done in expensive labs with very expensive equipment and gets funded by grants. The Alvarez Hypothesis was possible because of cutting edge nuclear science that could identify the iridium. LIGO, and every space science and astronomical discovery are driven by really expensive ever growing scopes. In astronomy, chem, physics, molecular biology, paleontology, et al today you'll need an expensive lab and very large grants to do cutting edge work. At this point we're building colliders many kilometers in length. Modern Science is a product of well funded labs with staffing and equipment funded largely by research grants. There are cases in domains like macrobiology where individuals could do field work and find a new species, but the domains where the individual scientist is doing meaningful work are dwindling to none, and you should update your model of how modern science actually operates and is funded, it's interesting.
That points to a bad system, but one that has nothing to do with the modern world we live in, and which if applied by analogy will fit so poorly as to be a bad analogy.
Also the Church conducted scientific experiments regularly, since a number of priests through the ages were scientists, and it was a result of priests taking observations in later, better telescopes that updated the Church's position on heliocentrism.
That aside, individual priest carried out their experiments, not in the name of the Church. The Church had already leaned towards scholarship by this time (post-Bologna) and being the dominant component of all European cultures, it only made sense that it would attract people with interests in science and experiments. (By the time we had scientist-priests people didn't know why they were Christians, to begin with.) This is discounting roles in Church or its bodies—usually monasteries—that required a vow to dumbness.
What do you mean?
The real gatekeeping comes in what research gets grants and funding, but if you look into what's happening there, it's not comparable to the Church proscribing things - it mostly means that the DoD, petrochem, and a few industries have outsized influence on what research is done.