If we allow politicians to turn science into a dogma instead of an investigative process, progress will come to a grinding halt.
I've tried this (by sharing data) with friends and family members - it doesn't work/matter. They simply dismiss the data as "fake news" and shift to a new argument. This is why I classify it as "disingenuous skepticism".
Their actions (anecdotally) are driven by culture and politics, not data.
You are right, simply presenting people with data doesn't work. If it did, you would never see a fat nurse, or a doctor who smokes, or... the list is endless.
pumaontheprowl's suggestion is to design a (thought) experiment that involves the skeptic. There are various ways to do this, but history is literally built on this exact phenomena. Getting people to change their mind on an issue requires that you have a real conversation with them. You can't just shove data in their face and see "See, look dummy you're wrong I'm right!" There are all sorts of people that do this for a living: helping people leave cults/white supremacy groups/etc.
And remember, the way most people are finding out about this "shared data" is via the media that just got done spending 4 years accusing trump of being a russian agent. So, they're biased or they're liars. Hence the skepticism.
I'd argue 40% of societies response to this pandemic is culture and politics. 30% is media fear mongering. 20% is intellectual error and 10% is actually disease mitigation.
Requiring fully vaccinated individuals to wear masks is culture & politics.
Shutting down outdoor restaurants is culture and politics
Casting moral shame on people who get covid is culture and politics
Saying it is totally okay to go to a BLM rally while not being able to attend a football game is culture and politics.
Closing the birthday card isle in the grocery store is culture and politics.
The list goes on and on...
In fact the very core of lockdowns is premised in culture and politics and not science. Science doesn't create policy. Human values do. For example places that place more emphasis on living life instead of protecting life were probably the first to drop restrictions.... a perfectly valid stance.
Science doesn't tell us what to do. It is merely a process for testing hypothesis in an attempt to get to the "truth" of something. That's all it does. Science didn't tell us to lockdown, it doesn't tell us to require vaccinated individuals to wear a mask, it doesn't tell us to shut down our borders, or any of that. All those decisions were driven by culture and politics.
For flat earthers in particular this video sheds some light on why their way of thinking is harmful: https://www.youtube.com/watch?v=JTfhYyTuT44
To be extremely reductive, allowing false beliefs into one's web of beliefs corrupts it. This typically gives rise to a multitude of other false beliefs. Flat earthers almost never only believe one obviously false conspiracy theory. They become epistemically susceptible, and worse still, they tend to spread these awful ways of thinking with great zeal.
In the more general case, I recommend you learn about the ethics of belief. The Standford Encyclopedia of Philosophy page on the topic is a good starting point: https://plato.stanford.edu/entries/ethics-belief/
Maybe someone can come along and answer your question in a more succinct yet equally (or ideally more) convincing way. This request that you commit several hours to educating yourself is the best I can do at this time.
And what's wrong with arguments against democracy? We are in a thread explicitly discussing questioning anything and everything and talking about the virtues of being averse to dogma. What if democracy is not as good as it gets? What if there's an alternative that is not authoritarianism?