I do not understand this utter contempt for human experience (aka "anecdotal evidence") for it is human experience which leads us to question the world around us and this natural tendency is what birthed your precious science.
Although this is indeed suggesting a false dilemma (if you are against me, you are against yourself), it is valid to wonder why anecdotal evidence is not given more value.
We do not hold anecdotal information in contempt, indeed it is the first step of the scientific method (observation). Notice that the word evidence is right there in the term. It's just that this evidence has no value in determining the probability of a claim being true. The reason is that the human experience is prone to mistakes of perception, logic, and memory. Anecdotal evidence is likely to be contaminated by memory drift, feedback, confirmation bias, unknown factors, selective perception, misinterpretation, exaggeration, delusion, ect ect. Most people are not aware of or likely to spot deception or hoax. At best, anecdotal information is prone to problems and impossible to test or verify. It is enough to start an investigation, to raise interest, and should not be ignored, but should also only be seen for what it is.
So why then does anecdotal information seem so convincing? Stories usually contain a lot of detail, especially after they have been told a few times. Details make a story seem credible, even though they may not be accurate. The person giving the account often seems honest and would have no motivation to deceive us, but we can't be sure they aren't deceiving themselves. Anecdotes told by authority figures seem especially credible, for no logical reason. There is also the bias of wanting to believe.
So should we listen to and examine closely anecdotal information? Absolutely! How else does a doctor decide to treat a patient if not by listening to his story? We simply must be careful that we are objective and consider all evidence, giving anecdotal evidence the proper weight it deserves.