It's someone strange that this idea–of everyone independently evaluating everything they read or hear–is so popular, considering it's entirely impossible to do even for just a fraction of the information you consume every day and rely on.
It doesn't even work in principle: as long as you are limited to receiving information over the internet, it's impossible to verify any of it from first principle. The results of some automatic processes can maybe use cryptography for verification. But any real-world fact relies on people observing it and providing you with texts/photos/videos: there's no end-to-end process to definitely proof that Angela Merkel is the chancellor of Germany. You have to either travel to Berlin and check for yourself, or you have to trust a single source completely, or you construct some most probable version of reality by aggregating many sources weighted by whatever trust you assign them.
"Having trust in..." is obviously the same as "appeal to authority". And if you belief you're not relying on that mechanism you're delusional.
Every time you trust Google Maps to get you to your destination, you do so because your experience with Maps specifically and Google more generally has taught you that this company and product can be trusted to a certain degree.
Every time you're more open to lending your car to your spouse than some random stranger, you're acting on trust.
Every time you return to your favourite restaurant, you're relying on the intuition that past behaviour is predictive of future behaviour. That's yet another formulation for "trust".
Do you believe there's a virus going around right now? How do you know? Are you just counting the number of sources saying so? Because in that case I can probably find more anonymous twitter accounts denying it. Do you consider a story in the New York Times to carry more weight than @justin1256 on Twitter? That's trust.
When you get a vaccination, you're relying on the combined authority of your doctor, and your country's licensing scheme for physicians, and the FDA and the company producing the vaccine.
What does "thinking for yourself" mean in that context? Reading a few papers online doesn't qualitatively change anything: you're just distributing your trust more widely. In practice, it typically means giving yourself license to find whatever study or YouTube video best conforms to your priors. Those people currently setting 5G towers on fire in Britain are excellent at "thinking for themselves" and "consuming a wide variety of sources".
Not relying on some sort of authority implies proving information from first principle. How do you go about it? Are you going to run your own clinical trial? That would cost billions, and expose dozens or hundreds of people to whatever danger you're trying to evaluate, defeating the purpose. It would probably also involve more work than you could do yourself, so you'd need to trust some collaborators. Foiled again!
I also don't get what's so bad about trust? If you want to define it, it's making predictions (about some person's or institution's honesty and competence) based on past experiences. Is there any difference between extending someone (limited) trust after a good experience to, say, jumping headfirst into a pool after you've swam in it to check its depth? Or cuddling a dog you know to be friendly? Yes, people can change (so can dogs), which is why trust isn't binary, and needs continuous updating. Which is why it's so fortunate that nature has evolved a whole range of emotions, from love to friendship to sympathy to fear, to keep track of this elusive quantity called trust.