That's just good practice to question things and do your research. There's nothing wrong with being overly critical if in the end you can still look at and react to the information objectively.
Again I disagree with silencing any doubters especially in scientific/research communities. I just expect standards; and I don't think that's to much to ask for. I can't stand when someone wants to present evidence, and it's a 1 off result out of 1000, so obviously it's the only result that matters because it fits the narrative. Another beauty is looking at the most basic data and ignoring key parts of the data to again fit the narrative.
A problem I often encounter is that people do not understand what even constitutes evidence. Peer reviewed journals, randomised control trials and mountains of data do not and should not have to justify themselves against anecdotes and YouTube videos. Your belief that vaccines are poisonous (for example) because some random conspiracy nut wrote it on their blog is not equivalent to literally billions of data points that say the opposite. "Do your own research" often doesn't work for this reason - because people don't understand what constitutes good research.
Now, I understand the scientific process, peer reviewing and scientific literature because I was educated in and work in a scientific field. Many people are not, and no disrespect to them - there are many paths to walk in life. But if you do not have the knowledge or skills to interpret scientific data, then "Do your own research" is bad advice. Your opinion which you reached through an evening of Googling does not hold the same value as the opinion of an expert working 20 years in the field with an abundance of data behind them.
I completely agree with you, however, that you should be allowed to hold and voice whatever opinions you like, just don't expect anyone to take you seriously or even listen to you if your opinions are stupid.