Hydroxychloroquine, sold as Plaquenil, is a very safe drug. It has been in use with many malaria, lupus and rheumatoid arthritis patients for decades. The question is whether it is effective against COVID-19. It might help to take a step back and look at terminology:
"Anecdotal evidence," in science, means evidence of an outcome not covered in a repeatable trial. It doesn't mean "out of left field," and the casual use of anecdotal evidence to indicate something with less credibility than a rumor isn't the same thing. Chloroquine is a toxic substance used to treat drug-resistant malaria cases, as a preferred alternative to letting the patient die. It is also used in fish tank cleaners. Not the same thing as hydroxychloroquine.
Medicine is the most primitive of our sciences, and one reason is that it is part science, part art. When science fails, the physician is left with art. When chemoradiation failed with my bladder cancer, my oncologist and I discussed alternatives. He had read the preliminary literature on a drug used for other purposes which he felt might work on my cancer; I read the material and concurred. I was recently discharged as cancer-free. Sometimes, when science cannot give a physician the answers he wants to have, the physician has to turn to art. What has worked under what circumstances with what kinds of patients with what symptoms in the past, and is there evidence of success in situations similar to those involving me and this patient? If the choices are between near-certain death and using a safe pill that will probably work, a physician's choice is pretty clear. That does not mean the physician believes that the drug has been proven effective to the gold standard of success in multiple repeatable trials. Thousands of physicians around the world treating hundreds of thousands of patients are prescribing hydroxychloroquine wearing their "artist" hat, not their "scientist" hat.
In the 1980s, after medical school, I was practicing psychiatry in Europe and became aware that manic-depressive patients, now called bipolar, were getting relief from use of anticonvulsants. I read the limited literature, talked with pharmacists and neurologists, and concluded that anticonvulsants were worth a try, because the alternative was antipsychotics, all of which had serious, bad side-effects. I explained my rationale to my patients, and prescribed the safest, oldest, anticonvulsants I could find for some bipolar patients. Today, anticonvulsants are proven to work in patients with bipolar disorder. The oath I took begins "First, do no harm." It does not begin, "First, do nothing."