Religions for many is what psychologist would call an "anchoring heuristic." What that means is that it is the anchor for their world view. A starting point or filter to understand most everything else. In the past "why does it rain?" "why do people get sick?" "why do people speak different languages?" "why are their earthquakes?" all had their answer with God, The Bible, or whatever version of holy scripture or belief one had been raised with or adopted. Now it's widely understood why it rains, what causes most illnesses, and how language has evolved through our sciences, such that we can predict rain, heal many diseases, and see the evolution of languages across time, as well as understand why some places have more earthquakes because of plate tectonics, which was ridiculed when first presented as a theory.
UFOs are the one of the biggest of mysteries that we don't understand well: and one that, like disease and earthquakes is scary at times. Just like people looked to the bible as clear evidence of disease and earthquakes being God's judgement with The Plagues of Egypt in Exodus, David's Census and the Plague in 2 Samuel 24, Miriam's Leprosy in Numbers 12 and Uzziah's in 2 Chronicles 26 and understood Earthquakes in Isaiah 29 and Ezekiel 38 as signs of God, some people will be tempted to look at NHI (non-human intelligence) and try to fit it into a biblical/religious narrative.
However, what we've found is that just like with the things already mentioned science has done a far suprerior job understanding the material world despite often intense religious pushback and cognitive dissonance. Galileo was sentence to house arrest until he died for supporting what Copernicus figured out with observation and math that the Sun, not the earth, was the center of the solar system (though at the time they thought it was the universe). It says in the bible in Ecclesiastes 1:5: "The sun rises and the sun sets, and hurries back to where it rises." And in Psalm 93 93:1: "The world is firmly established; it cannot be moved." Passages like these made it very clear to many that it was heretical and against the divine word of God that the earth could was not the center of all creation and moving around the sun. Another examples is the age of the earth with famously James Ussher trying to calculate the age of the world through biblical genealogies and came with the the earth being created in 4004 BC. now we know we radioactive dating that it's closer to 4.54 billion years old. There are others, but the last one I will mention is the creation of species. Gensis describes species being created according to their "kinds." Because of this most people, particularly relgious people, didn't believe the ideas first put forward by Darwin that species evolve. However, evidence has been built up over time to show the links between species that even the Catholic Church and many others accept evolution, and we routinly go to hospitals for treatments, and everyone believes the sun is the center of the solar sytem which is not even the center of the Milky Way, which is 1 of billions of galaxies.
Smallpox, Polio, Measles, Tetanus, Rabies, and others are all curable or preventable with moden science. Vaccines aren't always perfect, but compared to you or your child getting Polio or getting Rabies (which is otherwise 100% fatal), I'll take them.
When it comes to UFOs I think we are seeing a slow acceptance by some that there indeed may be other intelligence life in the universe. The Catholic Church as been open to the possibility of alien life (see below), and many other religions have either long-believed or have started to accept the possibility.
Science May Do Better, But it Doesn't a Free Pass Either
Like is says above science doesn't get a pass either. While Science has cured diseases, launed satellites, built airplanes, harness electricity and built the internet. Scientist have sometimes gotten so confident that they KNOW what reality is that they too have been unyielding to better data and have persecuted people for thinking things that have ultimately proven to be right. In the 1840s is was discovered by Ignaz Semmelweis that doctors washing their hands before delivering babies caused fewer deaths. Doctors pushed back and his ideas were ridiculed. I already mentioned plate tectonics before and how the idea was considered absurd by the geologists at the time, as well as scientist saying meteorites couldn't exist because "there are no rocks in space." There are other examples, some I mentioned already, like germ-theory replacing Miasma, the idea of the 4 elements (earth, water, fire, and air), the concept of Ether, The rejection of now accepted neuroplasticity, and there will be more to come.
The biggest challenge will be what will the NHI tell us about the world that many still use scripture or quickly outdated science to understand. For example If NHI made contact with people in the 1600s and told us that microorganisms caused most sickness it would be shocking, and many people might be in disbelief claiming that it's clear that disease is from God, the devil, sin, bad air, etc... What if aliens told us in 1300 that Humans were cousins to apes? Or told us in 1400 that stars are just other suns and we orbit ours just like there are other planets orbiting their suns? All of these revelations would have been such ontological shock and so distressing you might get a surge in end-time and death cults (which have happened with almost all of the big moments in history), demonic theory, or even a rejection of all science. What will we get told? We can only take guess and many have... That we are seeded here, that our model of physics is wrong in fundamental ways, that consciousness is not something unique to humans or even generated by your brain (it just receives/'filters it), that the afterlife exists but is not even close to what many people thought? That there are other "dimensions?" Who knows, but we need to realize that our understanding of reality, whether through religion, or science has often been wrong to some no matter how confident we've been. Some people think this distressing shock will be too much for humanity to collectivly handle and that's why there has been a persiting cover-up.